11 (c) Is the integer (447836)9 divisible Dy6. Working modulo 9 or 11, find the missing digits in the calculations below-(a) 51840- 273581 = 1418243x040.(b) 2x99561 = [[3(523 +x)]².(c) 2784x =x 5569.(d) 512 1x53125 = 1000000000.7. Establish the following divisibility criteria:(a) An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8.(b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 2(c) An integer is divisible by 4 if and only if the number formed by its tens and units[Hint: If n is even, then 10n = 1, 105n+1 = 10, 105n+2 = 100 (mod 1001); if n isdigits is divisible by 4.[Hint: 10% =0 (mod 4) for k > 2.](d) An integer is divisible by 5 if and only if its units digit is 0 or 5.8. For any integer a, show that a² - a +7 ends in one of the digits 3, 7, or 9.9. Find the remainder when 44444444 is divided by 9.[Hint: Observe that 23 = -1 (mod 9).]10. Prove that no integer whose digits add up to 15 can be a square or a cube.[Hint: For any a, a' = 0, 1, or 8 (mod 9).]11. Assuming that 495 divides 273x49y5, obtain the digits x and y.12. Determine the last three digits of the number 7999.[Hint: 74" = (1+ 400)" = 1+400n (mod 1000).]13. If t, denotes the nth triangular number, show that tn+2k = tn (mod k); hence, tn and tn+20must have the sɛme last digit.14. For any n > 1. prove that there exists a prime with at least n of its digits equa! to 0.[Hint: Consider the arithmetic progression 10"+Ik +1 for k = 1, 2, ....]15. Find the values of n 1 for which 1! +2! +3! + • · ·+ n! is a perfect square.[Hint: Problem 2(a).]16. Show that 2" divides an integer N if and only if 2" divides the number made up of thelast n digits ofN.[Hint: 10k = 2 5k = 0 (mod 2") for k > n.]17. Let N = am10m + +a210 + a¡10+ do, where 0 < ak <9, be the decimal expan-sion of a positive integer N.(a) Prove that 7,11, and 13 all divide N if and only if 7, 11, and 13 divide the integerM = (100a2 + 10a1 + ao) - (100a5 + 10a4 + a3)+ (100ag + 1Oa7 + a6) - .odd, then 103n =-1, 103n+1 = -10, 103n+2 =-100 (mod 1001).1(b) Prove that 6 divides N if and only if 6 divides the integerM = ao+4ai + 4a2 + +4am8. Without performing the divisions, detouby 7, 11, and 13

Name: 11 (c) Is the integer (447836)9 divisible Dy6. Working modulo 9 or 11,
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Description: 11 (c) Is the integer (447836)9 divisible Dy6. Working modulo 9 or 11, find the missing digits in the calculations below-(a) 51840- 273581 = 1418243x040.(b) 2x99561 = [[3(523 +x)]².(c) 2784x =x 5569.(d) 512 1x53125 = 1000000000.7. Establish the follo