Grade 12 - Exp. / Arithmetic and divisibility 56 exercises (100% corrected)

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1. Decomposition into products of prime factors E.8602 1 Name the prime numbers less than or equal to 20 . 2 Determine the decompositions into products of prime fac-tors of the following numbers : 126 ; 588 ; 5040 3 Deduce the decompositions into products of prime fac-tors of the following numbers : 5040 126 ; 126+588 E.5218 Determine the product of prime fac-tors decomposition of the following integers : a 8232 b 1750 c 1053 E.4288 k denotes a non-zero natural number. 1 Give the prime factor decomposition of 2008 . 2 Determine, explaining the method chosen, the smallest value of the natural number k for which k 6 is a multiple of 2008 . E.6805 On admits that any non-zero natural number n can be uniquely written as the product of a power of 2 by an odd integer: n =2 α × k où ¸ is a natural number (possibly zero) and k an odd nat-ural number. The writing n =2 α × k is named decomposition of n . For example, here are the decompositions of the integers 9 and 120 : 9 = 2 0 × 9 ; 120 = 2 3 × 15 . 1 Give the decomposition of the integer 192 . 2 Let x and z be two non-zero natural numbers, whose decompositions are: x =2 α × k ; z =2 β × m Write the decomposition of the natural integers 2 · x 2 and z 2 . 3 By examining the exponent of 2 in the decomposition 2 · x 2 and in that of z 2 , show that there is no pair of non-zero natural integers ( x ; y ) such that : 2 · x 2 = z 2 E.3364 Consider the following two integers : A =72 ; B =135 1 Determine the prime factor product decomposition of the integers A and B . 2 a Using a choice tree, determine the set of divisors of A and the set of divisors of B . b Give the set of divisors common to A and B . E.3622 Let n be a natural number. Consider the integer A defined by: A = 2 3 × 3 n × 5 n 1 a Determine the number of divisors of the integer A in the following cases : n = 0 ; n = 1 ; n = 2 b Determine an expression as a function of n giving the number of divisors of the integer A . 2 How many divisors does the integer 6 075 000 admit? 2. Parity E.3332 1 Intuitively complete the following two double entry ta-bles : + Pair Impair Pair Impair × Pair Impair Pair Impair 2 Noting the following two characterizations : If n ∈ Z is even, there exists k ∈ Z such that : n = 2 · k If n ∈ Z is odd, there exists k ∈ Z such that : n = 2 · k + 1 Answer the following questions : a Demonstrate that the sum of two odd integers is even. b Demonstrate that the product of two odd integers is odd. E.5035 Show that the sum of the squares of two consecutive integers is an odd integer E.3334 Consider the expression B = n 2 − 1 for n ∈ N ∗ : 1 Demonstrate that for n even, B is odd. 2 Demonstrate that for n an odd integer strictly greater than 1, B is even and divisible by 8. E.6773 Show that the sum of two consecu-tive odd integers is an integer multiple of 4 . E.3488 Let p and q be two natural numbers. Let’s study the relationship : p 2 − 2 · q 2 = 1 1 Find two integers p and q verifying the previous relation-ship and such that : 1 p 4 ; 1 q 4 2 It is assumed that the integers p and q verify the rela-tionship sought : a Show that the integer p is odd. b Deduce that the integer q is even. https://chingmath.fr chapExoCorrec/8602 sacados/8602 chapExoCorrec/5218 sacados/5218 chapExoCorrec/4288 sacados/4288 chapExoCorrec/6805 sacados/6805 chapExoCorrec/3364 sacados/3364 chapExoCorrec/3622 sacados/3622 chapExoCorrec/3332 sacados/3332 chapExoCorrec/5035 sacados/5035 chapExoCorrec/3334 sacados/3334 chapExoCorrec/6773 sacados/6773 chapExoCorrec/3488 sacados/3488
3. Some arithmetic problems E.3333 Determine the sum of all multiples of 11 between 100 and 400 . (we can use an arithmetic sequence) E.5726 Show that the sum of three consec-utive integers is divisible by 3 . E.8604 1 For k a non-zero natural number develop the expression : A = 2 · k +1 2 − 1 . 2 Justify that A is a multiple of 8 . E.6770 Recall that, for any non-zero natural number n , we have : 1 + 2 + · · · + n = n · n + 1 2 1 Show that the integer 1+2+ ··· + n is the square of an in-teger if, and only if, there exists a natural number p such that n 2 + n − 2 · p 2 =0 . 2 Deduce that the integer 1+2+ ··· + n is the square of an integer if, and only if, there exists a natural number p such that (2 n +1) 2 − 8 p 2 =1 4. Euclidean division in Z E.8603 1 Which of the proposed equalities represents the Eu-clidean division of − 25 by 3 : a − 25 = ( − 9) × 3 + 2 b − 25 = ( − 8) × 3 + ( − 1) 2 Determine the Euclidean divisions of the following num-bers by 4 : a − 31 b − 19 5. Euclidean division E.5728 From the equality: 456 164= 65 164 × 7+16 , deduce the Euclidean division of 456 164 by 7 . E.3327 1 Without the calculator, perform the Euclidean division of 372 by 15 . 2 With the calculator, determine the Euclidean division of 37 852 by 23 . 3 Use the following equalities to answer the following ques-tions : 2 456 = 17 × 143 + 25 ; 32 247 143 = 225 + 72 143 ; 516 43 = 12 674 24 = 28 + 1 12 ; 5 460 63 = 85 + 5 3 ; 345 = 27 × 13 − 6 a Determine the Euclidean division of 2 456 by 17 . b Give the Euclidean division of 32 247 by 143 . c Determine the remainder of the Euclidean division of 516 by 43 . d Determine the Euclidean division of 674 by 24 . e Deduce the Euclidean division of 5 460 by 63 . f Determine the Euclidean division of 345 by 13 . E.5840 Consider the function f ex-tracted from an algorithm whose call is made by passing it two natural numbers a and b as arguments : Function f(a,b) c ← 0 As long as a>b c ← c+1 a ← a − b End As long as Renvoyer ( c ; a) 1 Give the pair returned by the function f when called with the arguemnts a =13 and b =4 . In a step-by-step execution of the call to the function f , we will indicate the values of the variables a , b , c during this call. 2 What does the couple returned when calling the function ff represent? E.4696 The year 2012 was a leap year and 1 er January 2012 was a Sunday. 1 a How many days separate 1 er January 2012 and 20 January 2012 . b Give the Euclidean division of 19 by 7 . c What day of the week was 20 January 2012 . 2 Determine the day of the week of 25 March 2012 . https://chingmath.fr chapExoCorrec/3333 sacados/3333 chapExoCorrec/5726 sacados/5726 chapExoCorrec/8604 sacados/8604 chapExoCorrec/6770 sacados/6770 Extrait Asie Juin 2015 chapExoCorrec/8603 sacados/8603 chapExoCorrec/5728 sacados/5728 chapExoCorrec/3327 sacados/3327 chapExoCorrec/5840 sacados/5840 chapExoCorrec/4696 sacados/4696
E.6965 A credit card number is of the form : a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 a 13 a 14 a 15 c où a 1 , a 2 , . . . , a 15 and c are numbers between 0 and 9 . The first fifteen digits contain information on the card type, bank and bank account number. c is the number validation key. This number is calculated from the other fifteen. The function below validates the con-formity of a given card number. Fonction f( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 , a 10 , a 11 , a 12 , a 13 , a 14 , a 15 , a 16 , a 17 ,c) i ← 0 p ← 0 r ← 0 For k ranging from 0 to 7 r ← the remainder of the Euclidean division of 2 · a 2k+1 by 9 i ← i+r End For For k from 1 to 7 p ← p+a 2k End For s ← i+p+c If s is a multiple of 10 then : Return 1 Otherwise Return 0 End If Consider the following card number: 5635 4002 9561 3411 1 Complete the table below to obtain the final value of the variable I . k 0 1 2 3 4 5 6 7 a 2 k +1 2 a 2 k +1 r i 2 Justify that the card number 5635 4002 9561 3411 is cor-rect. 3 This card’s number is modified by changing the first two digits. The first digit (initially 5 ) is changed to 6 . What must be the second digit a for the resulting card number 6 a 35 4002 9561 3411 to remain correct? 6. Operation on Euclidean division E.3330 1 Determine the Euclidean division of 1 038 by 17. 2 By studying the square (61 × 17+1) 2 , determine the re-mainder of the Euclidean division of 1 038 2 by 17. 3 Deduce a conjecture about the remainder, for any natu-ral number n , the Euclidean division of 1 038 n by 17. E.6768 Let n be a relative integer. Consider the following two properties : P ˇ the remainder of the Euclidean division of n by 5 is 1 ı P ˇ the remainder of the Euclidean division of n by 4 is 3 ı 1 a Let n be a natural number verifying the property P . Determine the remainder of the Euclidean division of n − 11 by 5 . b Let n be a natural integer verifying the property P . Justify that the integer n − 11 is a multiple of 4 . 2 Without justification, give three integers verifying the two properties P and P . E.5979 n is a natural integer. In the Euclidean division of n by 7 , n can be written : n = 7 q + r où q and r are natural numbers. 1 What does r represent? What are the possible values for r ? 2 We divide n 2 by 7 . What are the possible values for r ? 3 Deduce that if 7 divides n 2 then 7 divides n . 4 n and m are two natural numbers. We divide n 2 + m 2 by 7 . What are the possible leftovers? 5 Deduce that if 7 divides n 2 + m 2 then 7 divides n and m . 7. Divisor and multiple E.3339 ¸ and ˛ represent two integers ; con-sider the following four sentences : 1 ¸ is a multiple of ˛ 2 ¸ is a multiple ˛ 3 ¸ is a divisor ˛ 4 ¸ has divisor ˛ The sentences above are equivalent two by two; find the equiv-alent sentences. E.5036 1 a Determine the remainder of the Euclidean division of 10 2 by 3 . b Using reasoning by recurrence, show that for any nat-ural integer n , the remainder of 10 n by Euclidean di-vision by 3 is 1 . 2 Justify that the integer 4 × 10 n − 1 is, for any natural num-ber n , divisible by 3 . https://chingmath.fr chapExoCorrec/6965 sacados/6965 Extrait Liban Juin 2017 chapExoCorrec/3330 sacados/3330 chapExoCorrec/6768 sacados/6768 chapExoCorrec/5979 sacados/5979 chapExoCorrec/3339 sacados/3339 chapExoCorrec/5036 sacados/5036
Débutdelafrise E.1821 For any natural number n , we pose : A ( n )= n 2 − n +2007 . The aim of the exercise is to study the divisibility of the inte-gers A ( n ) by 2 and by 3 . This exercise consists of two independent questions. 1 a Give the prime factor product decomposition of the integer A (1) equal to 2007. b Let n be a natural number. Show that : ˇ If n is divisi-ble by 3, then A ( n ) is divisible by 3 ı. c Is the reciprocal of this last statement true? Justify. 2 a Verify that, whatever the natural number n , we have : ( n + 1) 2 − ( n + 1) + 2007 = ( n 2 − n + 2007) + 2 · n b Consider any natural number n . Show that : ˇ If A ( n ) is odd, then A ( n +1) is impair ı. c Is the following statement true or false? Justify. ˇ There exists at least one natural number n such that A ( n ) is divisible by 2 ı. E.3360 For any relative integer n , consider the integer A n defined by: A n =2 n 2 +2 n +8 Justify that the integer A n is divisible by 4 for any relative integer n ∈ Z E.6783 Let n be a relative integer. Establish the following property: 2 n +3 is a multiple of 7 = ⇒ 3 n +1 is a multiple of 7 . 8. Problem and remainder of Euclidean division E.5701 A frieze is made up of squares, triangles, circles and trapezoids in regular succession. These elements are successively painted white, striped or black. The beginning of the frieze is shown below : 1 Give the characteristics of 113 ième element of this frieze. 2 What is the element following 113 ième element and hav-ing the same characteristics. E.5702 A coding system transforms any letter in a text into another, rendering the text unreadable. It does this by numbering the letters of the alphabet, start-ing with 0 . A transformation on the integer then changes the letter. Here’s the correspondence table for this coding : A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 C F I L O R U X A D G What transformation was used on integers? 9. Handling the remainder of Euclidean division E.3357 In this exercise, we are interested in Euclidean division by 4, and more specifically in the remain-der of division by 4: 1 Let x and y be two integers whose remainder when di-vided by 4 is 2 and 3 respectively. a Give an expression characterizing the Euclidean divi-sion of x by 4 as well as for the Euclidean division of y by 4 . b Establish that the remainder of the division ( x + y ) by 4 is equal to 1 . c Establish that the remainder of the division x · y by 4 is equal to 2 2 Let x and y be two integers ; we denote r x and r y as the remainders of the division of x and y by 4, respectively. Consider the two tables below : r x r y 0 1 2 3 0 1 2 3  r x + y r x r y 0 1 2 3 0 1 2 3  r x · y We note r x + y and r x · y the respective remainders by di-viding the integers ( x + y ) et ( x · y ) . a Enter the results obtained in question 1 in these two tables. b Complete these two tables ˇ intuitively ı. https://chingmath.fr chapExoCorrec/1821 sacados/1821 Antilles-Guyane - juin 2007 - 6 points chapExoCorrec/3360 sacados/3360 chapExoCorrec/6783 sacados/6783 chapExoCorrec/5701 sacados/5701 Débutdelafrise chapExoCorrec/5702 sacados/5702 chapExoCorrec/3357 sacados/3357
E.3331 We give the Euclidean division of 195 695 by 3 : 195 695 = 65 231 × 3 + 2 1 Justify that the remainder of the Euclidean division of 195 695 × 2 by 3 is 1 . 2 Determine the remainder of the Euclidean division of 195 695 × 3 by 3 . 3 Let r n be the remainder of the Euclidean division of 195 695 × n by 3. Complete the following table from memory: n 1 2 3 4 5 6 7 r n E.8605 Euclidean division of the integer 7 17 by 4 has remainder 3 . The following questions will be an-swered, justifying your answer. 1 Determine the remainder of the Euclian division of 7 17 by 2 . 2 Determine the remainder of Euclidean division 7 34 by 4 . E.6785 We give the following two Euclidean divisions by 5 : 5474 = 1094 × 5 + 4 ; 5487 = 1097 × 5 + 2 Without using the calculator and without calculating a prod-uct, show that the integer 5474 2 +5487 2 is divisible by 5 . Your entire approach must be present on your copy. E.6772 n is a natural integer. In the Euclidean division of n by 7 , n can be written : n = 7 · q + r où q and r are natural numbers. 1 What does r represent? What are the possible values for r ? 2 We divide n 2 by 7 . What are the possible leftovers? 3 Deduce that if 7 divides n 2 then 7 divides n . 10. Exhaustive reasoning E.5727 Determine the set of couples ( a ; b ) of natural numbers verifying the equality: a 2 − b 2 = 21 E.4999 Consider the integer A defined by: A = m × 4 · n + 1 où m; n ∈ N . We seek to determine the values of m and n achieving equality A =45 . 1 By studying the relationship 45= m × (4 · n +1) , give a set of possible values of m . 2 Deduce the set of couples ( m ; n ) realizing A =45 . E.3335 1 Determine the value of a and b two relative integers such that, for any relative integer n , we have : 4 n + 1 n + 1 = a + b n + 1 2 Deduce the values of n for which 4 n +1 n +1 is an integer. E.3358 Let a and b be two natural numbers. 1 Demonstrate that : a 2 · b − a · b 2 = 9 = ⇒ a · b divides 9 and a − b divides 9. 2 Deduce the pairs of solutions ( a ; b ) such that : a 2 · b − a · b 2 = 9 . E.3359 Determine the set of natural num-bers n such that the quotient of the Euclidean division of n by 5 is equal to the remainder by the same division. E.8609 Consider the integer B defined by: B = 2 · m + 3 · 2 · n où m; n ∈ N Determine the set of couples ( m ; n ) of integers verifying the relation B =70 . E.8607 Determine, if they exist, the values of n ∈ Z so that the fraction 6 n +9 2 n +1 has an integer value : E.8608 Determine, if they exist, the values of n ∈ Z so that the fraction 9 − 6 n 3 n − 4 has an integer value. E.3490 For any relative integer n different from 1 , consider the number: A n = 2 n 2 − n − 11 n − 1 1 Determine the value of the relative integers a , b , c ver-ifying the following relation for any natural number n distinct from 1: A n = a · n + b + c n − 1 2 Deduce the values of n such that the integer A n is an integer. E.8606 Consider for any relative integer, the number B n defined by: B n = 2 n 2 − 3 n − 15 2 n + 3 Determine the values of n for which B n is a relative integer. E.5039 Determine the set of pairs ( a ; b ) of rela-tive integers verifying the equality: a 2 − b 2 = 11 11. Case disjunction reasoning https://chingmath.fr chapExoCorrec/3331 sacados/3331 chapExoCorrec/8605 sacados/8605 chapExoCorrec/6785 sacados/6785 chapExoCorrec/6772 sacados/6772 chapExoCorrec/5727 sacados/5727 chapExoCorrec/4999 sacados/4999 chapExoCorrec/3335 sacados/3335 chapExoCorrec/3358 sacados/3358 chapExoCorrec/3359 sacados/3359 chapExoCorrec/8609 sacados/8609 chapExoCorrec/8607 sacados/8607 chapExoCorrec/8608 sacados/8608 chapExoCorrec/3490 sacados/3490 chapExoCorrec/8606 sacados/8606 chapExoCorrec/5039 sacados/5039
E.6774 Show that the product of three con-secutive integers is divisible by 6 . To do this, consider the following three cases : The first of the integers is divisible by 3 . The first of the integers has a remainder of 1 by Euclidean division by 3 . The first of the integers has a remainder of 2 by Euclidean division by 3 . E.5730 Show that for any natural number n , the expression 3 n 2 + n +2 is divisible by 2 . E.3623 Let N be a natural, odd, non-prime integer. Assume that N = a 2 − b 2 où a and b are two natural numbers such that a>b . 1 Show that a and b do not have the same parity. 2 Show that N can be written as the product of two natural numbers p and q . 3 What is the parity of p and q ? 12. Reasoning by the absurd E.6769 In this exercise, we are in-terested in triplets of non-zero natural integers ( x ; y ; z ) such that : x 2 + y 2 = z 2 These triplets will be named ˇ Pythagorean triplets ı in refer-ence to the right-angled triangles whose sides they measure, and noted in abbreviated form ˇ TP ı. Thus 3 ; 4 ; 5 is a TP because : 3 2 + 4 2 = 9 + 16 = 25 = 5 2 1 Demonstrate that, if ( x ; y ; z ) is a TP , and p a non-zero natural number, then the triplet ( p · x ; p · y ; p · z ) is also a TP . 2 Demonstrate that, if ( x ; y ; z ) is a TP , then the natural numbers x , y and z cannot all be three odd. E.6775 Consider the following equa-tion in Z : ( E ) : a 2 + 9 = 2 40 1 Show that if a exists then a is odd. (we can use reasoning by the absurd) 2 a Complete the following table : r 1 3 5 7 r 2 Rest of r 2 par the division euclidienne by 8 b Euclidean division of a by 8 yields the existence of a single pair ( q ; r ) : a = 8 · q + r ; 0 r< 8 Justify that if the integer a is odd, then the expression a 2 +9 is not divisible by 8 . c Deduce that the equation ( E ) admits no solution. E.6933 In this exercise, we’re interested in triplets of non-zero natural integers ( x ; y ; z ) such as : x 2 + y 2 = z 2 These triplets will be named ˇ Pythagorean triplets ı in refer-ence to the right-angled triangles whose sides they measure, and noted in abbreviated form ˇ TP ı. Thus, 2 ; 3 ; 4 is a TP because : 3 2 +4 2 =5 2 Part A : generalities 1 Demonstrate that, if ( x ; y ; z ) is a TP, and p a non-zero natural number, then the triplet ( px ; py ; pz ) is also a TP. 2 Show that, if ( x ; y ; z ) is a TP, then the natural numbers x , y and z cannot all be odd threes. 3 For this question, we assume that any non-zero natural integer n can be uniquely written as the product of a power of 2 by an odd integer: n = 2 α × k où ¸ is a natural number (possibly zero) and k an odd natural number. Writing n =2 α × k is named decomposition of n . For example, here are the decompositions of the integers 9 and 120 : 9 = 2 0 × 9 ; 120 = 2 3 × 15 a Give the decomposition of the integer 192 . b Let x and z be two non-zero natural numbers, whose decompositions are: x = 2 α × k ; z = 2 β × m Write the decomposition of the natural integers 2 x 2 and z 2 . c By examining the exponent of 2 in the decomposition of 2 x 2 and in that of z 2 , show that there is no pair of non-zero natural integers ( x ; y ) such that 2 x 2 = z 2 . We admit that the question A 3 establishes that the three natural numbers x , y and z are two distinct. Since, moreover, the natural integers x , y play a symmetrical role, in the fol-lowing, for any TP ( x ; y ; z ) , the three natural numbers x , y and z will be arranged in the following order : x <y <z Part B; searching for Pythaogric triplets containing the integer 2015 1 Decompose the integer 2015 into a product of prime fac-tors and then, using the TP given in the preamble, de-termine a TP of the form x ; y ; 2015 . 2 We admit that, for any natural number n : 2 n + 1 2 + 2 n 2 + 2 n 2 = 2 n 2 + 2 n + 1 2 Determine a TP of the form (2015 ; y ; z ) 3 a Noting that 403 2 =169 × 961 , determine a pair of non-zero natural numbers ( x ; z ) such that : z 2 − x 2 =403 2 with x< 403 b Deduce a TP of the form ( x ; 2015 ; z ) . https://chingmath.fr chapExoCorrec/6774 sacados/6774 chapExoCorrec/5730 sacados/5730 chapExoCorrec/3623 sacados/3623 Centres etrangers Juin 2005 chapExoCorrec/6769 sacados/6769 chapExoCorrec/6775 sacados/6775 chapExoCorrec/6933 sacados/6933
13. Contrapositive reasoning E.6784 Let n be a natural number. Prove the following assertion : n 2 is odd = ⇒ n is odd. 14. Unclassified financial years E.5820 Consider the function f extracted from an algorithm: Function f(A) X ← A As long as X 26 X ← X − 26 End of as long as Return X 1 What is the value returned by the function f when called with integer 3 as argument? 2 What is the value returned by the function f when called with the integer 55 as argument? 3 For any integer entered, what does the value returned by this function represent? E.5835 Consider the function f below, taken from an algorithm, où Ent A N denotes the integer part of A N . Function f(A) N ← 1 As long as N √ A Si A N − Ent A N =0 Then ( X ; Y) ← N ; A N End if N ← N+1 End As long as 1 By calling the function f with the value 12 for the argu-ment A , what values will be assigned to the ( X ; Y) pair of variables when this program is called. 2 What are the values assigned to the variable X when this program is run? https://chingmath.fr chapExoCorrec/6784 sacados/6784 chapExoCorrec/5820 sacados/5820 Extrait du Bac Antilles-Guyanes Septembre 2013 chapExoCorrec/5835 sacados/5835 Extrait d'Antilles-Guyane Juin 2012