4. Suppose that a positive integer is written in decimal notation as n = azak-1a₂ª1ª0 where 0 ≤ a;< 9. Prove that n is divisible by 11 if and only if the alternating sum of its digits a-a; + ... + (−1)kakis divisible by 11.

Here, we use theory of congruences.

Answered: 4. Suppose that a positive integer is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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9. Prove that n is odd, then n² - 1 is divisible by 8.
Assume that the sum of the digits of a number, n, in base 10, equals to 16. Example: n= 97 is one such number since 9+7=16, and n=4444 is another What are the possible values for the sum of the digits, S, of 2n? Example: If n=97 then 2n= 2*97=194 so S=1+9+4=14. If n=4444 then 2*4444= 8888 and S=4*8= 32. Find all possible values for S, the sum of the digits of 2n, and prove that no other values are possible for S.
10. Prove that n³-n is divisible by 6 for every integer n.
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If [5 - -8 A = [611] 0 " then A- 1 =
A certain website wants you to build a 6-character password from the letters a through i, the numbers 0−9, or the symbols @, #, or &. Clearly, the order of the characters in the password matters(a) How many passwords are there in total?(b) How many passwords are there that consist of distinct characters?(c) How many passwords have the first two characters be letters, the middle two character be one of the symbols and the last two characters be numbers where each character does not have to be distinct?
Assume that the sum of the digits of a number, n, in base 10, equals to 16. Example: n= 97 is one such number since 9+7=16, and n=4444 is another What are the possible values for the sum of the digits, S, of 2n? Example: If n=97 then 2n= 2*97=194 so S=1+9+4=14. If n=4444 then 2*4444= 8888 and S=4*8= 32. Find all possible values for S, the sum of the digits of 2n, and prove that no other values are possible for S.
5. Prove that a natural number (am-1am-2 ·a1a0)10 is divisible by 9 if and only if the sum of the digits an-1 + an-2 +.+ ai + ao is divisible by 9. n-.
3. There are seven students who always compare their CS test grades. They discovered that either the sum of two grades or the difference of two grades was always divisible by 10. Use the Pigeonhole Principle to show that for any seven positive integers, either the sum of two or the difference of two is divisible by 10.
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3.) Which positive integers have an odd number of positive divisors?
6. Let a be a number written in base 10 as x = ao + a110 +..a, 10" where 0 < a; < 10. (a) Prove that 2 divides a if and only if 2 divides ao. (b) Prove that 4 divides a if and only if 4 divides ao +2a1. (c) Prove that 8 divides a if and only if 8 divides ao + 2a1 + 4a2. (d) Prove that 5 divides a if and only if 5 divides ao. (e) Prove that 9 divides a if and only if 9 divides the sum ao + ..an of its digits. (f) Prove that 3 divides a if and only if 3 divides the sum ao + ...an of its digits. (g) Prove that 11 divides a if and only if 11 divides the alternating sum ao – a1 + az · (h) Come up with a rule for dividing by 7 and prove that it works.

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4. Suppose that a positive integer is written in decimal notation as n = akak-1 a2a1ao where 0