Math 310: Homework 8 (Ch 19-20, 24) – due Wednesday, 12/3
1. (V.1) Prove, for any positive integer n, that
7 divides 6n+ 1 if and only if nis odd.
2. (V.2) Prove that, for all integers aand b,a2+b2≡0, 1, 2, 4, or 5 modulo 8.
Deduce that there do not exist integers aand bsuch that a2+b2= 12345790.
3. (V.3) Suppose that a positive integer is written as n=akak−1...a2a1a0where 0≤
ai≤9. Prove that nis divisible by 9 if and only if the sum of its digits, ak+ak−1+
... +a1+a0is divisible by 9.
(hint: this problem is similar to problem 19.3)
4. (V.9) Solve the following linear congruences
(i) 3x≡15 mod 18
(ii) 3x≡16 mod 18
(iii) 4x≡16 mod 18
(iv) 4x≡14 mod 18
5. Solve 23x≡16 mod 107
The next two questions need results from Chapter 24:
6. What is the last digit of 21000?
7. Find the reminders for:
(i) 29! divided by 31
(ii) 18! divided by 23
(iii) 18! divided by 437
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