Elementary Number Theory
Cam McLeman
October 26, 2006
One of number theory’s claims to fame is the unusual ease with which one can
pose exceedingly difficult problems, and the enormity of the full toolkit needed
to tackle all of these problems. Fortunately, the good news is that solving
a Putnam problem is rarely about having memorized the applicable theorem.
Instead, there are a select few elementary results/topics that can cover a wide
range of possible questions:
•Modular arithmetic.
•Euler’s theorem (in particular Fermat’s Little Theorem).
•Prime factorization, gcd’s, and divisibility.
Example 1.Show that the sequence
11,111,1111,11111,· · ·
contains no perfect squares.
Example 2.Prove that the fraction
21n+ 4
14n+ 3
is irreducible for every positive integer n.
Example 3.What are the last two digits of 332006?
Example 4.Suppose that the number of prime divisors of a positive integer n
is a prime number pwhich does not divide n. Show that nis one more than a
multiple of p.
Problem 1.If 2n+1 and 3n+ 1 are both perfect squares, show that nis divisible
by 40.
Problem 2.How many trailing zeros are at the end of the decimal expansion of
150
72 ?
Problem 3.Find all positive integers dsuch that ddivides both n2+ 1 and
(n+ 1)2+ 1 for some n.
Problem 4.For any prime p, prove that every prime divisor of 2p−1 is at least
p.
1
