Grade 9 / gcd and lcm 57 exercises (including 56 corrected)

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×501×515×5225×503×5115×5275×509×5145×52225×502×5110×5250×506×5130×52150×5018×5190×52450×504×5120×52100×5012×5160×52300×5036×51180×52900×30×31×32×30×31×32×30×31×32202122 ChingQuizz : 6 exercises available for Quizz assessment : 1. Set of divisors E.4997 Consider the integer A defined by: A = 2 2 × 5 Among the following integers, name the factors of A : 2 ; 2 2 ; 2 3 ; 2 × 5 ; 2 × 5 2 Justify your answers. E.9150 Consider the integer A worth 60 . 1 Determine the value of the integers m , n , p positive ver-ifying the equality: 60=2 m × 3 n × 5 p 2 Of the following numbers, name the factors of A : 2 ; 2 2 ; 2 3 ; 3 × 5 2 ; 3 2 × 5 E.5000 1 Determine the integers m , n , p and q verifying the equal-ity: 28=2 m × 3 n × 5 p × 7 q 2 Using the previous question, list the six factors of the integer 28 . E.9152 1 Determine the prime factor product decomposition of the integer 30 . 2 Deduce the list of eight factors of the integer 30 . E.9153 1 Determine the prime factor product decomposition of the integer 90 . 2 Derive the list of twelve factors of the integer 90 . E.10674 1 Give the prime factor product decomposition of the inte-ger 168 . 2 The integer 168 has 16 divisors. Give in 6 as a product of product factors. E.10675 1 Give the prime factor product decomposition of the inte-ger 448 . 2 The integer 448 has 14 divisors. Give in 6 as a product of product factors. E.9859 Determine the smallest odd positive integer that has three different prime factors. Ex-plain your reasoning. E.9151 1 Give the prime factor product decomposition of the inte-ger 90 . 2 The tree below represents all numbers written in the form 2 i × 3 j × 5 k where the integers i , j , k have values ranging from 0 to 2 . Give the set of factors of the number 90 . Hint: parts of branches not representing factors of 180 will be hatched. 2. Common Divisors and Prime Factorization E.11661 1 Donner la décomposition en produit de facteur premiers des entiers 20 et 150 2 a Est ce que 2 est un diviseur commun à 20 et 150 ? b Même question pour 2 2 . 3 Est ce que 3 est un diviseur domun à 20 et 150 ? 4 Quel est le plus grand diviseur commun à 20 et 150 ? E.11662 1 Donner la composition en produit de facteurs premiers des entiers 40 et 84 . 2 Donner le plus grand diviseur commun à 40 et 84 . 3. Greatest common divisor: list of divisors E.10651 Definition: Let a and b be two integers. We call ˇ the greatest common divisor of a and b ı, noted GCD( a , b ) , the largest integer that divides both the integer a and the integer b . https://chingmath.fr chapExoCorrec/4997 sacados/4997 chapExoCorrec/9150 sacados/9150 chapExoCorrec/5000 sacados/5000 chapExoCorrec/9152 sacados/9152 chapExoCorrec/9153 sacados/9153 chapExoCorrec/10674 sacados/10674 chapExoCorrec/10675 sacados/10675 chapExoCorrec/9859 sacados/9859 chapExoCorrec/9151 sacados/9151 ×501×515×5225×503×5115×5275×509×5145×52225×502×5110×5250×506×5130×52150×5018×5190×52450×504×5120×52100×5012×5160×52300×5036×51180×52900×30×31×32×30×31×32×30×31×32202122 chapExoCorrec/11661 sacados/11661 chapExoCorrec/11662 sacados/11662 chapExoCorrec/10651 sacados/10651
Example: consider a =36 and b =60 . divisors of 36 : 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 36 divisors of 60 : 1 , 2 , 3 , 4 , 5 , 6 , 10 , 12 , 15 , 20 , 30 , 60 The common divisors of a and b are: 1 , 2 , 3 , 4 , 6 , 12 . We can deduce that : GCD(a,b)=12 1 Give the 6 divisors of the integer 12 . 2 Give the 8 divisors of the integer 42 . 3 Deduce the value of the ˇ greatest common divisor ı of the integers 12 and 42 . E.10670 1 a Give the 6 divisors of 28 . b Give the 6 divisors of 98 . 2 Deduce the value of PGCD (28; 98) . E.10654 1 a Give the 8 divisors of 54 . b Give the 6 divisors of 243 . 2 Deduce the value of PGCD (54; 243) . E.11284 1 a Give the 4 divisors of the integer 14 . b Give the 4 divisors of the integer 21 . 2 What is the greatest common divisor of the integers 14 and 21 ? E.11285 1 a Give the 4 divisors of the integer 10 . b Give the 4 divisors of the integer 27 . 2 What is the greatest common divisor of the integers 10 and 27 ? E.11286 1 a Give the 6 divisors of the integer 28 . b Give the 4 divisors of the integer 35 . 2 What is the greatest common divisor of the integers 28 and 35 ? E.11298 1 a Give the 8 divisors of 54 . b Give the 6 divisors of 63 . 2 Deduce the value of PGCD (54 ; 63) . E.11329 1 a Give the 8 divisors of 42 . b Give the 8 divisors of 70 . 2 Deduce the value of PGCD (42; 70) . 4. Greatest common divisor: list of divisors and problems E.11299 1 Give the prime factorization of 70 and 42 . 2 A cabinetmaker has a wooden board measuring 70 cm by 42 cm . He wants to cut the board into identical rectangles mea-suring 12 cm 2 , with dimensions in centimeters that are whole numbers. In addition, he wants to use the entire board. a Give the dimensions of these wooden rectangles. b How many rectangles can he make? E.11331 1 a Give the 8 divisors of 128 b Give the 8 divisors of 40 2 A cabinetmaker has a wooden board measuring 128 cm × 40 cm . He wants to : cut squares of the same size and as large as possible ; their dimensions are whole numbers of centimeters ; he wants to use the entire board How many squares will he cut from this board? 5. Use of the greatest common divisor E.9325 A teacher is organizing a field trip to the Futuroscope for his ninth graders. He wants to divide the 126 boys and 90 girls into groups. He wants each group to have the same number of girls and the same number of boys. 1 Decompose the numbers 126 and 90 into products of prime factors. 2 Find all integers that divide both the numbers 126 and 90 . 3 Deduce the largest number of groups the teacher will be able to form. How many girls and boys will there be in each group then? https://chingmath.fr chapExoCorrec/10670 sacados/10670 chapExoCorrec/10654 sacados/10654 chapExoCorrec/11284 sacados/11284 chapExoCorrec/11285 sacados/11285 chapExoCorrec/11286 sacados/11286 chapExoCorrec/11298 sacados/11298 chapExoCorrec/11329 sacados/11329 chapExoCorrec/11299 sacados/11299 chapExoCorrec/11331 sacados/11331 chapExoCorrec/9325 sacados/9325
E.10658 The club president wants to offer small, all-identical gift bags containing stickers and flags with the club logo. She has bought 330 stickers and 132 flags and wants to use them all. She wants exactly the same num-ber of stickers in each bag and exactly the same number of flags in each bag. 1 Why isn’t it possible to make 15 bags? 2 a Decompose 330 and 132 into product of prime fac-tors. b Deduce the greatest number of bags the chairwoman will be able to make. c In this case, how many stickers and flags will she put in each bag? E.10659 1 Anne and Jean bought 630 pink sugared almonds and 810 white sugared almonds, which they put in a bag. We as-sume that the sugared almonds are indistinguishable to the touch. a How many sugared almonds did Anne and Jean buy in total? b Anne randomly picks a sugared almond from the bag. What is the probability that she will get a white sug-ared almond? 2 They are using these sugared almonds to make wedding favors, so that : the number of pink sugared almonds is the same in each box; the number of white sugared almonds is the same in each box; all the sugared almonds are used. a Can they make 21 boxes? b Factorize 630 and 810 into prime factors. c Deduce the maximum number of boxes that Anne and Jean can make. Then give the composition of each box. E.10729 1 a Determine the prime factor product decomposition of the integers 144 and 168 . b Deduce the value of the greatest common divisor of 144 and 168 . 2 A construction company has 144 kg of limestone gravel and 168 kg of marble gravel. It wishes to create bags of limestone gravel and marble gravel with the following constraints : all gravel must be used in the composition of the bags ; each bag must have the same weight ; the company wants to make the biggest bags possible. Determine the number of bags of limestone gravel and the number of bags of marble gravel that this company will be able to make. E.9327 1 Decompose the numbers 162 and 108 into products of prime factors. 2 Give at least one common factor to the numbers 162 and 108 strictly greater than 10 (three answers are possible) 3 A snack bar sells trays consisting of egg rolls and sa-moussa. The cook has prepared 162 nems and 108 samossas. In each tray: the number of egg rolls must be the same. the number of samossas must be the same. All egg rolls and all samossas must be used. a Can the cook make 36 trays? b What is the maximum number of trays he will be able to make? c In this case, how many egg rolls and samossas will be in each tray? E.9323 The captain of a ship has a treasure trove of 69 diamonds, 1150 pearls, and 4140 gold coins. 1 Decompose 69 , 1150 and 4140 into products of prime factors. 2 The captain divides the treasure equally among the sailors. How many sailors are there knowing that all the coins, pearls, and diamonds have been distributed? E.10652 Two florists have the same stock: 300 irises and 675 roses. They each want to use all these flowers, but have their own ideas about how to use them : The florist A wants to make iris bouquets and rose bou-quets each with the same number of flowers. He also wants these bouquets to have as many flowers as possi-ble. The florist B wishes to make the maximum number of bouquets all identical where each bouquet contains irises and roses. 1 Determine the number of iris bouquets and the number of rose bouquets made by the florist A . 2 Determine the number of bouquets of flowers (and their composition) made by the florist B . E.10656 A salesman has a stock of 252 bot-tles of tiare perfume and 315 monoi soap. He wants to sell all this stock by making as many ˇ Souvenirs de Polynésie ı boxes as possible so that : the number of tiare perfume bottles in each box is the same ; the number of monoi soaps be the same in each box; all bottles and soaps be used. Find the number of boxes to be prepared and the composition of each. The assessment of this question will take account of observa-tions and research steps, even if incomplete ; show them on the copy. https://chingmath.fr chapExoCorrec/10658 sacados/10658 chapExoCorrec/10659 sacados/10659 Martinique Juillet 2024 chapExoCorrec/10729 sacados/10729 chapExoCorrec/9327 sacados/9327 chapExoCorrec/9323 sacados/9323 chapExoCorrec/10652 sacados/10652 chapExoCorrec/10656 sacados/10656
E.10697 A jeweler owns 135 rubies and 162 diamonds. He wishes to manufacture rings featuring rubies and diamonds with the following constraints : each ring includes the same number of rubies and the same number of diamonds. he wants to make as many rings as possible. he wishes to use all the rubies and diamonds. 1 Determine the number of rings made by this jeweler. 2 How many rubies and diamonds does each of these rings include? E.10698 A jeweler has 135 rubies and 162 diamonds. He wants to make rings with rubies and rings with diamonds, subject to the following constraints : Each ring must have the same number of gemstones. He wants the rings to have as many gemstones as possi-ble. He wants to use all the rubies and diamonds. 1 Determine the number of gemstones in each of these rings. 2 How many ruby rings and diamond rings will the jeweler make? E.11742 Une entreprise de plomberie dispose de 416 joins en caoutchouc et 128 joins en fibre. 1 a Déterminer la décomposition en produit de facteurs premiers des entiers 416 et 128 . b Donner le PGCD des entiers 416 et 128 . 2 Elle souhaite créer des sachets de joins en caoutchouc et des sachets de joins en fibre tels que : Tous les sachets ont lés mêmes nombres de joins. Tous les joins doivent être utilisés. Il souhaite que chaque sachet est le plus grand nombre possible de joins. Combien de joins contient chacun des sachets? 3 Finalement, elle décide de créer des sachets qui contien-nent des joins en fibre et des joins en caoutchouc tels que : Tous les sachets contiennent le même nombre de joins en caoutchouc et le même nombre de joins en fibre. Tous les joins doivent être utilisés. Il souhaite créer le plus grand nombre possible de sa-chets. Combien de sachets va-t-elle composer? 6. Numbers that are prime to each other E.9146 Definition: let a and b be two non-zero integers. We say that the integers a and b are prime to each other if the integer 1 is the only common divisor to a and b . Note: as an equivalent definition, we have the proposition : Two integers a and b are prime if, and only if, PGCD ( a;b ) = 1 . Proposition: let a and b be two non-zero integers. If the integers a and b admit no prime integer as a common divisor then the integers a and b are prime to each other. Example: let’s consider : a =24 ; b =35 We have the decompositions into products of prime integers : a = 2 3 × 3 ; b = 5 × 7 The integers a and b admit no prime integer as a common divisor. We deduce that the integers 24 and 35 are prime to each other. Which of the following pairs of integers form a pair of prime integers : a (32 ; 81) b (28 ; 21) c (17 ; 36) E.10695 1 Determine the prime factor product decomposition of the integers 208 and 585 . 2 Are the integers 208 and 585 prime to each other? Justify your answer. E.10696 1 Determine the prime factorization of the integers 136 and 315 . 2 Are the integers 136 and 315 relatively prime? Justify your answer. E.2134 Specify whether the following statements are true or false. Justify. 1 3 25 is a decimal number. 2 The integers 570 and 795 are prime to each other. E.9148 What number satisfies the following three conditions? I am a number written 15 x where x is a number between 0 and 9 . I am first with the integer 126 . I am not a prime integer. Hint: here are the prime integers less than 200 : 2 ; 3 ; 5 ; 7 ; 11 ; 13 ; 17 ; 19 ; 23 ; 29 ; 31 ; 37 ; 41 ; 43 ; 47 ; 53 ; 59 ; 61 ; 67 ; 71 ; 73 ; 79 ; 83 ; 89 ; 97 ; 101 ; 103 ; 107 ; 109 ; 113 ; 127 ; 131 ; 137 ; 139 ; 149 ; 151 ; 157 ; 163 ; 167 ; 173 ; 179 ; 181 ; 191 ; 193 ; 197 ; 199 7. Irreducible fractions https://chingmath.fr chapExoCorrec/10697 sacados/10697 chapExoCorrec/10698 sacados/10698 sacados/11742 chapExoCorrec/9146 sacados/9146 chapExoCorrec/10695 sacados/10695 chapExoCorrec/10696 sacados/10696 chapExoCorrec/2134 sacados/2134 Extrait France - Septembre 2007 chapExoCorrec/9148 sacados/9148
0123456789101112 E.9147 Definition: let a and b be two integers where b =0 . The fraction a b is said to be irreducible if the integers a and b are prime to each other. Which of the fractions below are given in their irreducible form : a 15 27 b 14 27 c 36 15 d 49 64 E.8005 Useful data. The beginning of the ordered list of prime numbers is : 2 ; 3 ; 5 ; 7 ; 11 ; 13 ; 17 ; 19 ; 23 ; 29 1 Decompose 140 and 870 into product of prime numbers. 2 Deduce the irreducible form of the fraction 140 870 . E.8006 For the question posed, only one of the three proposed answers is correct. Specify which one and justify your choice. The irreducible form of the fraction 882 1 134 is : a 14 9 b 63 81 c 7 9 E.648 1 a Give the list of divisors of the eight divisors of the integer 30 and of the eight divisors of the integer 24 . b What are the common divisors of the integers 30 and 24 ? 2 Write the fraction 30 24 as an irreducible fraction 3 Perform the following calculation: 30 24 − 3 4 E.8011 Determine the irreducible form of the fractions below : a 96 36 b 102 120 c 220 260 E.5680 1 Determine the prime factor product decomposition of the following integers : 18 ; 30 ; 45 2 Using the previous question, simplify the following frac-tions :: a 30 45 b 18 30 c 18 45 E.8007 Determine the irreducible form of the fractions below : ca 315 225 b 616 924 c 1 512 630 E.5001 1 Determine the decomposition of the integers below into the product of prime factors : a 108 b 432 c 588 2 Using the previous question, simplify the following frac-tions : a 108 432 b 588 108 c 432 588 8. Prime numbers and probabilities E.8000 Consider a game consisting of a ro-tating board and a ball. Shown op-posite, this board has 13 squares numbered from 0 to 12 . The ball is tossed onto the board, eventually stopping at random on a numbered square. The ball has the same probability of stopping on each square. 1 What is the probability that the ball will stop on the square numbered 8 ? 2 What is the probability that the number of the square on which the ball stops is an odd number? 3 What is the probability that the number of the square on which the ball stops is a prime number? E.8001 A bag contains 20 balls each with the same probability of being drawn. These 20 balls are numbered from 1 to 20 . One ball is drawn at random from the bag. All results will be given as irreducible fractions. 1 What is the probability of drawing the numbered ball 13 ? 2 What is the probability of drawing an even numbered ball? 3 Is it more likely to get a ball with a number that is a multiple of 4 than to get a ball with a number that is a factor of 4 ? 4 What is the probability of drawing a ball with a number that is a prime? 9. Smallest common multiple E.10661 Definition: Let a and b be two integers. We call the least common multiple , noted PPCM ( a;b ) , the small-est integer that is both a multiple of a and a multiple of b . https://chingmath.fr chapExoCorrec/9147 sacados/9147 chapExoCorrec/8005 sacados/8005 chapExoCorrec/8006 sacados/8006 chapExoCorrec/648 sacados/648 chapExoCorrec/8011 sacados/8011 chapExoCorrec/5680 sacados/5680 chapExoCorrec/8007 sacados/8007 chapExoCorrec/5001 sacados/5001 chapExoCorrec/8000 sacados/8000 0123456789101112 chapExoCorrec/8001 sacados/8001 chapExoCorrec/10661 sacados/10661
R1R2 R1R2R3 Circuit 1Ex1Ex2Ex3Ex4Ex5Départ / ArrivéeCircuit 2Ex1Ex2Ex3Ex4Ex5Ex6Ex7Ex8Ex9Ex10Départ / Arrivée Example: for a =12 and b =15 . multiples of 12: 12, 24, 36, 48, 60 , 72, 84, 96, 108, 120 . . . multiples of 15 : 15, 30, 45, 60 , 75, 90, 105, 120 , 135, 150. . . The value of PPCM (12 ; 15) is 60 . 1 a Give the ten multiples of the number 16 . b Give the ten multiples of the number 12 . 2 Deduce the value of PPCM (12 ; 16) . E.10703 1 a Give the first five multiples of the integer 12 . b Give the first five multiples of the integer 16 . 2 Give the Least Common Multiple of the integers 12 and 16 . 10. Least Common Multiple and Factorization E.11663 1 Déterminer la décomposition en produit de facteurs pre- miers de 20 et de 75 . 2 En déduire la valeur de PPCM (20 ; 75) . 11. Using the smallest common multiple E.10676 Consider the gear below made up of gears R 1 , R 2 composed of 12 and 21 identical teeth on the two gears respectively. Initially, the arrows marked on the wheels R 1 and R 2 face up-wards. As soon as we start turning the wheel R 1 , the arrows turn in opposite directions. How many complete turns of the gear R 1 must be made at least for the two arrows to return to their initial positions : pointing upwards? E.10660 Consider the gear system shown below, consisting of gears R 1 , R 2 , and R 3 , which have 18, 10, and 20 identical teeth, respectively. Initially, the arrows indicated on the gear wheels are pointing upwards. As soon as the gear R 1 begins to turn, the arrows follow the movement of the wheels. How many complete turns of gear R 1 must be made at a minimum for these three arrows to return to their starting position? E.10655 A sports coach prepares two training circuits containing several cardio and strength train-ing exercises : one circuit starts with exercise 1 and ends by returning to exercise 1 ; circuit 1 contains five exercises. Each exercise lasts 40 seconds and must be followed by 16 seconds of rest to allow time to move on to the next exercise; circuit 2 contains ten exercises. Each exercise lasts 30 seconds and must be followed by 5 seconds of rest to allow time to move on to the next exercise. 1 Show that circuit 1 takes 280 seconds and circuit 2 takes 350 seconds. 2 Give the prime factorization of 280 and 350 . 3 A training session consists of several laps of the same circuit. When the coach blows the whistle, Camille begins a train-ing session on circuit 1 and Dominique on circuit 2 . a Explain why, when 2800 seconds have elapsed since the whistle blew, Camille is back at the start of circuit 1 . Specify where Dominique is on the circuit 2 when 2800 seconds have elapsed b After the whistle, how long does it take Camille and Dominique to meet at the same time for the first time at the start of their circuit? Express this time in min-utes and seconds? https://chingmath.fr chapExoCorrec/10703 sacados/10703 chapExoCorrec/11663 sacados/11663 chapExoCorrec/10676 sacados/10676 R1R2 chapExoCorrec/10660 sacados/10660 R1R2R3 chapExoCorrec/10655 sacados/10655 Circuit 1Ex1Ex2Ex3Ex4Ex5Départ / ArrivéeCircuit 2Ex1Ex2Ex3Ex4Ex5Ex6Ex7Ex8Ex9Ex10Départ / Arrivée
ABCD90m70m E.6103 We admit the following proposition : Proposition: let k be an integer and n 1 and n 2 be two non-zero integers. If the remainder of the Euclidean division of k by n 1 and the remainder of the Euclidean division of k by n 2 have the same value r then the integer k has the value : k = PPCM ( n 1 ;n 2 ) + r Let n be an integer such that : the remainder of n by 168 has the value 3 ; the remainder of n by 63 has the value 3 . Determine the value of n . E.10894 We accept the following proposi-tion : Proposition: n 1 , n 2 , r 1 , r 2 four non-zero integers such that : r 1 <n 1 ; r 2 <n 2 ; n 1 − r 1 = n 2 − r 2 We then note : r = n 1 − r 1 = n 2 − r 2 Let k be an integer. If the Euclidean division of k by n 1 is r 1 and if the Euclidean division of k by n 2 is r 2 , then we have the value of k k = PPCM ( n 1 ;n 2 ) − r Let n be an integer such that : the remainder of n divided by 27 is 18 ; the remainder of n divided by 72 is 63 . Determine the value of n . E.11673 Marc et Jim, deux amateurs de course à pied, s’entraînent sur une piste d’athlétisme dont la longueur du tour mesure 400 m . Marc fait un temps moyen de 2 min-utes par tour. Marc commence son entraînement par un échauffement d’une longueur d’un kilomètre. 1 Combien de temps durera l’échauffement de Marc ? 2 Quelle est la vitesse moyenne de course de Marc en km = h ? À la fin de l’échauffement, Marc et Jim décident de com-mencer leur course au même point de départ A et vont ef-fectuer un certain nombre de tours. Jim a un temps moyen de 1 minute et 40 secondes par tour. Le schéma ci-dessous représente la piste d’athlétisme de Marc et Jim constituée de deux segments [ AB ] et [ CD ] et de deux demi-cercles de diamètre [ AD ] et [ BC ] . (Le schéma n’est pas à l’échelle et les longueurs indiquées sont arrondies à l’unité.) ABCD est un rectangle tel que AB =90 m et AD =70 m 3 Calculer le temps qu’il faudra pour qu’ils se retrouvent ensemble, au même moment, et pour la première fois au point A . Puis déterminer combien de tours de piste cela représentera pour chacun d’entre eux. Indication : toute trace de recherche, même non aboutie, devra apparaître sur la copie. Elle sera prise en compte dans l’évaluation. 12. Unclassified exercises E.10976 1 Determine the prime factorization of the integer 504 . 2 Determine the GCD of the integers 216 and 126 . 3 Give the reduced form of the fraction 126 216 4 Give the prime factorization of the integer 42 and the integer 78 . Deduce the prime factorization of 42 2 × 78 . 5 A florist has 42 roses and 78 tulips. He wants to use all his flowers to make bouquets of roses and bouquets of tulips, each with the same number of flowers. In ad-dition, he wants the number of flowers per bouquet to be as large as possible. How many roses and how many bouquets of tulips will he make? Hint: Each of your answers must be justified by showing the steps of your calculations and reasoning. https://chingmath.fr chapExoCorrec/6103 sacados/6103 chapExoCorrec/10894 sacados/10894 chapExoCorrec/11673 sacados/11673 ABCD90m70m chapExoCorrec/10976 sacados/10976