(1) How many positive integers n are such that n is divisible by 8 and n+1 is divisible by 25?
(2) How many positive integers n are such that n is divisible by 21 and n+1 is divisible by 165?
(3) Find all integers n such that n is divisible by 9, n+1 is divisible by 25 and n+2 is divisible by 4.
Answer
infinitely many, none, n = 774 mod 900
Solution
(1) We need n = 0 mod 8 and -1 mod 25. Hence n = 24 mod 200.
(2) We need n = 0 mod 21, n = -1 mod 165. But 3 divides 165, so we require n = 0 mod 3 and 2 mod 3, which is impossible.
(3) We need n = 0 mod 9, -1 mod 25, 2 mod 4. Hence n = 99 mod 225, and 2 mod 4, so n = 774 mod 900
Thanks to Suat Namli
© John Scholes
jscholes@kalva.demon.co.uk
6 March 2004
Last corrected/updated 6 Mar 04