Prime Numbers, The Greatest Common Divisor Worksheet - Math 289 Page 8
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8Exercise 11. **** Suppose that p, q, r are positive integers which gcd(p, q) = gcd(p, r) =
gcd(q, r) = 1. Show that if N ≥ 3pqr, then any N × N floor can be tiled with tiles of sizes
p × p, q × q and r × r.
Exercise 12. ** Suppose that m and n are positive integers with gcd(m, n) = 1. Show that
mn
m
n is the largest integer that is not of the form mx + ny with x, y ≥ 0.
n
Exercise 13. ** Show that if 2
1 is a prime number, then n is a prime number.
n
Exercise 14. ** Show that if 2
+ 1 is a prime number, then n = 0 or n is a power of 2.
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