Grade 9 / Arithmetic 61 exercises (100% corrected)

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ChingQuizz : 4 exercises available for Quizz assessment : 1. Reminders about powers E.11261 Perform the following calcula-tions : a 5 − 2 × 6 2 b 2 × (30 − 5 2 ) c 4 + 18 − 12 2 E.11262 Proposal: To multiply two powers of the same number, add the exponents. Example : 8 2 × 8 3 = 8 × 8 × 8 × 8 × 8 = 8 5 To divide two powers of the same number, add the exponents. Example : 7 5 7 3 = 7 × 7 × 7 \× 7 × 7 7 × 7 × 7 = 7 × 7 × 7 × 7 × 7 7 × 7 × 7 = 7 2 Simplify the following expressions : a 3 5 × 3 8 b 7 10 7 6 c 6 8 × 6 4 f 6 7 6 4 E.11263 Simplify the writing of the follow-ing expressions : a 3 2 × 3 4 b 5 8 × 5 7 c 3 × 3 4 d 3 5 3 2 e 8 3 8 2 f 7 12 7 5 E.11270 Definition: For any non-zero number a ( a =0 ) and for any positive integer n , we note : 1 a n = a − n a 0 = 1 ; a 1 = a Examples: 5 3 5 7 = 1 5 4 = 5 − 4 ; 7 − 2 × 7 6 = 1 7 2 × 7 6 = 7 6 7 2 = 7 6 − 2 = 7 4 Complete the following calculations : a 5 4 5 7 = 5 ... = 1 5 ... b 7 4 × 7 − 10 = 7 ... = 1 7 ... c 3 13 3 20 = 3 ... = 1 3 ... d 8 − 5 × 8 2 = 8 ... = 1 8 ... E.11328 Simplify the following expressions : a 3 5 × 3 3 × 3 8 b 7 12 7 5 c 5 7 × 5 4 5 9 d 9 2 × 9 3 9 5 × 9 10 E.11264 Simplify the writing of expressions : a 10 × 10 − 4 10 − 8 b 10 5 × 10 − 4 10 − 3 c 10 − 12 × 10 8 10 4 E.11265 Simplify the writing of expressions : a 10 3 10 − 2 4 b 10 16 10 2 8 c 10 5 10 7 2 d 10 3 − 2 E.11266 Proposition: for a a positive integer, m and n two strictly positive integers : a n × a m = a n + m ; a n a m = a n − m ; a n m = a n × m Simplify the following powers : a 5 2 × 5 5 b 7 4 × 7 − 7 c 5 × 5 − 4 d 3 5 × 3 2 4 e 8 5 × 8 − 3 × 8 − 2 f 5 20 × 5 − 9 3 2. Euclidean division E.11333 Perform the following Euclidean divisions : a 132 by 8 b 213 by 6 Note: the results will be written in the form : Dividende = Quotient × Diviseur + Reste E.9853 1 Complete the blanks below using integers : a 87 = : : : × 4 + 7 b 87 = : : : × 4 + 3 c 87 = : : : × 4 − 1 d 87 = 23 × 4 + : : : 2 Which of the following equalities represents the Eu-clidean division of 87 by 4 ? https://chingmath.fr chapExoCorrec/11261 sacados/11261 chapExoCorrec/11262 sacados/11262 chapExoCorrec/11263 sacados/11263 chapExoCorrec/11270 sacados/11270 chapExoCorrec/11328 sacados/11328 chapExoCorrec/11264 sacados/11264 chapExoCorrec/11265 sacados/11265 chapExoCorrec/11266 sacados/11266 chapExoCorrec/11333 sacados/11333 chapExoCorrec/9853 sacados/9853
E.2155 Which of the following equalities represents the Euclidean division of 375 by 14 ? a 375 = 25 × 14 + 25 b 375 = 26 × 14 + 11 c 375 = 27 × 14 − 3 E.9852 1 Complete the blanks below using integers : a 148 = : : : × 13 + 5 b 148 = : : : × 13 + 18 c 148 = : : : × 13 − 8 d 148 = 13 × 13 + : : : 2 Which of the following equalities represents the Eu- clidean division of 148 by 13 ? E.9851 For each question, determine the equality expressing the Euclidean division of a by b : a a = 370 ; b = 250 b a = 315 ; b = 16 c a = 1 254 ; b = 26 d a = 24 576 ; b = 134 E.9870 For each question, determine the equality expressing the Euclidean division of a by b : a a = 29 ; b = 29 b a = 65 ; b = 120 3. Divisors of a number E.9854 1 a Give the six factors of the number 12 . b Give all the factors of the number 27 . 2 What are the common factors of the integers 12 and 27 . E.2133 In this exercise, any evidence of research, even if incomplete, or of initative, even if unsuc-cessful, will be considered in the evaluation. ˇ The hidden number: I am a whole number between 100 and 400. I am even. I am divisible by 11. I also have 3 and 5 as factors. Qui suis-je? ı. Explain a process for finding the hidden number, and give its value. E.9149 1 Give the eight factors of the integer 30 and the eight factors of the integer 24 . 2 What are the common factors of the integers 30 and 24 ? E.4998 Complete the table below : Entier x 2 3 4 5 6 7 8 9 10 11 12 Nombre de diviseurs de x 2 2 3 E.10682 Which of the triplets of numbers below contains three divisors of the integer 144 : a 2 ; 9 ; 4 b 2 ; 3 ; 7 c 4 ; 5 ; 9 4. Integer parity E.10902 Complete the following two double-entry tables : + Even Odd Even Odd × Even Odd Even Odd E.10903 Complete the following sentences without justification using the words even , odd , any . a The sum of two even integers is an integer . . . . . . b The sum of two odd integers is an integer . . . . . . c The product of two odd integers is an integer . . . . . . d The product of an even integer by an odd integer is an integer . . . . . . E.10904 Without justification, say whether the following assertions are true, false or undecidable : 1 The sum of two odd integers is an even integer. 2 The product of an even integer and an odd integer is even. 3 The product of two consecutive integers is an even inte-ger. 4 The sum of five consecutive integers is a multiple of 5. https://chingmath.fr chapExoCorrec/2155 sacados/2155 chapExoCorrec/9852 sacados/9852 chapExoCorrec/9851 sacados/9851 chapExoCorrec/9870 sacados/9870 chapExoCorrec/9854 sacados/9854 chapExoCorrec/2133 sacados/2133 chapExoCorrec/9149 sacados/9149 chapExoCorrec/4998 sacados/4998 chapExoCorrec/10682 sacados/10682 chapExoCorrec/10902 sacados/10902 chapExoCorrec/10903 sacados/10903 chapExoCorrec/10904 sacados/10904
•Choisir un nombre•Calculer son carré•Ajouter6au résultat Choisir un nombreSoustraire 2Multiplier par4Elever au carréAjouter les deux nombres E.10872 Consider the following calculation program : Are the following statements true or false? Justify your answers and write down the steps for any cal-culations : 1 ˇ If we choose the number 2 3 , the result of the program is 58 9 . ı 2 ˇ If an integer is chosen, the result of the program is an odd integer. ı 3 ˇ The result of the program is always a positive number. ı 4 ˇ If the chosen number is an even integer, then the result is an even integer. ı E.10901 Consider the calculation pro-gram below : 1 ˇ If we choose the number 2 3 , the result of the pro-gram is 40 9 . ı 2 ˇ If an integer is chosen, the result of the program is an odd integer. ı 3 ˇ The result of the program is always a positive number. ı 4 ˇ If the chosen integer is an even integer then the result is an integer pair ı 5. Prime numbers E.8008 Definition: An integer a is said to be prime if it has exactly two divisors ( 1 and itself) In the list of numbers below, cross out the numbers that are not prime integers : 33 47 51 28 39 49 85 E.10445 Justify that each of the integers below is not a prime integer: 573 ; 1784 ; 1065 E.8003 Indicate and justify whether the following statement is true or false. Assertion: ˇ The number 231 is a number premier ı E.8004 In this exercise, one question is asked and only one of the four answers provided is correct. State the correct answer and justify your choice. The numbers 23 and 37 : a are first b are divisible by 3 . c have no common factors d are even. E.8009 Tell whether the statement below is true or false, justifying the answer. Assertion: for any positive integer n , the number 2 n +1 is a prime number. E.10714 Which of the following integers is a prime integer: 16 993 ; 16 984 ; 17 007 ; 16 983 ; 16 985 E.10722 Which of the following integers is a prime number: 17 401 ; 17 396 ; 17 391 ; 17 409 ; 17 395 E.8002 Say whether the statement, with justification, is true or false. Assertion : For all integers n between 2 and 9 , the integer 2 n − 1 is a prime. E.9855 1 Give three pairs of prime integers whose sum is a prime integer. 2 Consider a pair ( a ; b ) of prime integers such that a + b is a prime integer. Study the parity of the integers a and b . 6. Prime factorization (with 2, 3, 5, 7) E.9265 The number 588 can be de-composed as : 588=2 2 × 3 × 7 2 What are its prime factors? That is, numbers that are both primes and factors of 588 . https://chingmath.fr chapExoCorrec/10872 sacados/10872 •Choisir un nombre•Calculer son carré•Ajouter6au résultat chapExoCorrec/10901 sacados/10901 Choisir un nombreSoustraire 2Multiplier par4Elever au carréAjouter les deux nombres chapExoCorrec/8008 sacados/8008 chapExoCorrec/10445 sacados/10445 chapExoCorrec/8003 sacados/8003 chapExoCorrec/8004 sacados/8004 chapExoCorrec/8009 sacados/8009 chapExoCorrec/10714 sacados/10714 chapExoCorrec/10722 sacados/10722 chapExoCorrec/8002 sacados/8002 chapExoCorrec/9855 sacados/9855 chapExoCorrec/9265 sacados/9265
126263321371reétape2ndétape3eétape 180.........52......3 84 294 378 420 504 1080 E.9856 Example: to determine the decomposition of the integer 126 into a product of prime factors, we use the following algorithm which is schematized in the diagram below : We look for a prime factor of 126 : for example 2 . We obtain the equality: 126 = 2 × 63 We look for a prime factor of 63 : for example 3 . We obtain the equality: 126 = 2 × 3 × 21 We look for a prime factor of 21 : for example 3 . We obtain the equality: 126 = 2 × 3 × 3 × 7 We stop since 7 is also a prime number. Determine the product of prime factors decomposition of the integers below : a 45 b 140 c 196 E.5678 Complete the diagram below to ob-tain the prime factorization of the integer 180 . The diagram can be used as a guide : E.11297 Give the prime factorization of the integer 84 . The diagram can be used to help : E.10728 Give the prime factorization of the integer 294 The diagram can be used to help : E.10727 Give the product of prime factors decomposition of the integer 378 The diagram can be used as a guide : E.11295 Give the prime factorization of the integer 420 . The diagram can be used to help : E.11412 Give the prime factorization of the integer 504 : The diagram can be used to help : E.11296 Give the prime factorization of the integer 1080 . The diagram can be used to help : E.9858 1 Determine the prime factor decomposition of 27 000 000 . 2 What are its prime factors? 7. Prime factor decomposition https://chingmath.fr chapExoCorrec/9856 sacados/9856 126263321371reétape2ndétape3eétape chapExoCorrec/5678 sacados/5678 180.........52......3 chapExoCorrec/11297 sacados/11297 84 chapExoCorrec/10728 sacados/10728 294 chapExoCorrec/10727 sacados/10727 378 chapExoCorrec/11295 sacados/11295 420 chapExoCorrec/11412 sacados/11412 504 chapExoCorrec/11296 sacados/11296 1080 chapExoCorrec/9858 sacados/9858
891 1650 E.10723 Give the prime factorization of the integer 891 : The diagram can be used to help : E.10726 Give the prime factor product de-composition of the integer 1650 : The diagram can be used as a guide : 8. Operations on prime factor decompositions E.10692 Let a and b be two integers, in each of the cases below, give the prime factor product decomposi-tion of the integer a × b : a a = 2 3 × 3 × 5 2 et b = 3 5 × 5 b a = 3 4 × 5 × 7 2 et b = 2 × 3 3 E.11260 1 Give the prime factorization of the integers : 175 ; 108 2 Noting that 175 × 108=18 900 , give the prime factoriza- tion of 18 900 . E.8010 Determine the product of prime fac-tors decomposition of each of the numbers below : a 14 × 12 b 35 × 24 c 16 × 54 E.10701 1 Give the prime product decomposition of the integers 162 and 270 . 2 Deduce the prime product decomposition of the integer 162 × 270 . 9. Quotient of factorizations into prime factors E.10672 For each of the fractions below, give their irreducible expression : a 2 4 × 3 8 × 7 3 2 7 × 3 6 × 7 2 b 2 12 × 3 4 × 5 5 2 8 × 3 4 × 5 7 c 2 34 × 5 17 2 30 × 3 3 × 5 15 E.10699 Let a and b be two integers. For each question, give the prime factorization of the integer a b : a a = 2 3 × 3 × 5 3 et b = 2 2 × 3 × 5 b a = 2 4 × 3 4 × 7 2 et b = 2 × 3 3 E.10724 Consider the two integers : a = 2 6 × 3 × 5 2 × 7 2 ; b = 2 4 × 5 2 × 7 For each question, give the prime factorization of the calcula-tion : a a × b b a b E.10702 1 Give the prime product decompositions of the integers 2160 and 36 . 2 Deduce the prime product decomposition of the integer 2160 36 . E.11330 1 Give the prime factorization : a of the integer 364 b of the integer 378 2 Deduce the prime factorization of 364 × 378 . 3 Give the reduced form of the fraction 364 378 . E.11413 1 Give the prime factorization of the integers 180 and 378 . 2 Deduce the prime factorization of the integer 180 × 378 . 3 Deduce the reduced expression of the fraction 180 378 . E.10673 1 Give the prime factor product decomposition of the inte-gers 12 , 44 and 72 . 2 Deduce the irreducible expression of the fraction 12 × 44 72 . 10. Factorization of decompositions into prime factors https://chingmath.fr chapExoCorrec/10723 sacados/10723 891 chapExoCorrec/10726 sacados/10726 1650 chapExoCorrec/10692 sacados/10692 chapExoCorrec/11260 sacados/11260 chapExoCorrec/8010 sacados/8010 chapExoCorrec/10701 sacados/10701 chapExoCorrec/10672 sacados/10672 chapExoCorrec/10699 sacados/10699 chapExoCorrec/10724 sacados/10724 chapExoCorrec/10702 sacados/10702 chapExoCorrec/11330 sacados/11330 chapExoCorrec/11413 sacados/11413 chapExoCorrec/10673 sacados/10673
E.10704 Establish that the integers below are multiples of 5 : a 2 155 × 3 78 + 2 154 × 3 79 b 2 64 × 3 23 × 7 49 + 2 62 × 3 24 × 7 50 E.10725 Establish that the integer A is a multiple of 7 : 2 53 × 3 40 + 2 55 × 3 39 E.10705 Consider the integer a defined by: a = 4 17 × 3 27 × 5 17 + 2 31 × 9 13 × 5 19 Show that the integer a is a multiple of 7 . 11. Deepening: a little arithmetic E.9240 1 a Determine the prime factor product decomposition of 2 744 . b Derive the prime factor product decomposition of 2 744 2 . c Using this decomposition, find x such that : x 3 = 2 744 2 2 Let a and b be two integers greater than 2 such that a 3 = b 2 . a Calculate b when a =100 . b Determine two integers a and b greater than 2 and less than 10 that verify the equality a 3 = b 2 . E.9860 The sum of two multiples of 5 is always a multiple of 5 . https://chingmath.fr chapExoCorrec/10704 sacados/10704 chapExoCorrec/10725 sacados/10725 chapExoCorrec/10705 sacados/10705 chapExoCorrec/9240 sacados/9240 chapExoCorrec/9860 sacados/9860 Extrait France - Septembre 2007