7b,11 when17. Let N = am10" + . . + a2102 + a¡10+ ao, where 0 < ak <9, be the decimal expan-[Hint: If n is even, then 10n = 1, 103n+1 = 10, 103n+2 = 100 (mod 1001); if n is(a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer74ELEMENTARY NUMBER THEORY(b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of Nwritten in the base 9.(c) Is the integer (447836)9 divisible by 3 and 8?6. 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 3.(c) An integer is divisible by 4 if and only if the number formed by its tens and unitsdigits is divisible by 4.[Hint: 10k = 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 tn denotes the nth triangular number, show that t+2k = tn (mod k); hence, t, and t,420must have the same last digit.14. For any n > 1. prove that there exists a prime with at least n of its digits equal to 0.[Hint: Consider the arithmetic progression 10"+'k+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.]sion of a positive integer N.M = (100a2 + 10a1 + ao) - (100a5 + 10a4 + az)+ (100as + 10a7 + a6) –odd, then 103n =-1, 103+1 =-10, 103n+2 = -100 (mod 1001).1(b) Prove that 6 divides N if and only if 6 divides the integerM = ao + 4a1 +4a2 + +4am

Name: 7b,11 when17. Let N = am10" + . . + a2102 + a¡10+ ao, where 0 < ak <9,
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: 7b,11 when17. Let N = am10" + . . + a2102 + a¡10+ ao, where 0 < ak <9, be the decimal expan-[Hint: If n is even, then 10n = 1, 103n+1 = 10, 103n+2 = 100 (mod 1001); if n is(a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide