(b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

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7b,11
when
17. 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 integer
74
ELEMENTARY NUMBER THEORY
(b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of N
written 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 units
digits 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,420
must 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 the
last 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 integer
M = ao + 4a1 +4a2 + +4am
Transcribed Image Text:when 17. 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 integer 74 ELEMENTARY NUMBER THEORY (b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of N written 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 units digits 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,420 must 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 the last 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 integer M = ao + 4a1 +4a2 + +4am
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