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last digit of divisor

Source: 30-th Vietnamese Mathematical Olympiad 1992

February 17, 2007

number theory unsolvednumber theory

Problem Statement

For any positive integer

a

, denote

f(a)=|\{b\in\mathbb{N}| b|a

\text{and}

b\mod{10}\in\{1,9\}\}|

and

g(a)=|\{b\in\mathbb{N}| b|a

\text{and}

b\mod{10}\in\{3,7\}\}|

. Prove that

f(a)\geq g(a)\forall a\in\mathbb{N}