Worksheet on topic 4 - Remainders when you divide by powers Math 165 - 11/19
1. (VTRMC 5, 1979) Show, for all positive integers n = 1, 2, . . . that 14 divides 34n+2 +52n+1 .
2. It is a fact that if a and b are positive integers, and d = gcd(a, b) is the greatest common
divisor of a and b, then there are integers x and y so that ax + by = d.
Use this to prove that gcd(2a − 1, 2b − 1) = 2gcd(a,b) − 1.
[ First show that 2gcd(a,b) − 1 divides 2a − 1 and 2b − 1. Then prove that if m is a positive
integer and 2a ≡ 1 (mod m) and 2b ≡ 1 (mod m), then 2gcd(a,b) ≡ 1 (mod m).]
3. (VTRMC 1, 1981) The number 248 − 1 is divisible by what two numbers between 60 and
70?
20000
10
4. (Putnam A2, 1986) What is the units (i.e., rightmost) digit of b 10
100 +3 c. Here bxc is the
greatest integer ≤ x.
5. (VTRMC 2, 2010) For a positive integer n, define f1 (n) = n and then for i a positive
integer, define fi+1 (n) = fi (n)fi (n) . Determine f100 (75) mod 17 (i.e. determine the remainder
after dividing f100 (75) by 17, an integer between 0 and 16). Justify your answer.
6. (IMO 4, 1975) When 44444444 is written in decimal notation, the sum of its digits is A.
Let B be the sum of the digits of A. Find the sum of the digits in B. (A and B are written
in decimal notation.)
1
2
7. (VTRMC 4, 2012) Define f (n) for n a positive integer by f (1) = 3 and f (n + 1) = 3f (n) .
What are the last two digits of f (2012)?
8. (Putnam B5, 1997) Prove that for n ≥ 2,
2
2
.
.
..
..
n ≡ 22
n−1≡
22
(mod n).
(Another way of saying this is to define a1 = 2 and ak = 2ak−1 . Then the goal is to prove
that for each n ≥ 2, an ≡ an−1 (mod n).)