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Turkey NMO 2007 1st Round - P30 (Number Theory)

Source:

October 5, 2012

modular arithmetic

Problem Statement

Let

(a_n)_{n=1}^{\infty}

be an integer sequence such that

a_{n+48} \equiv a_n \pmod {35}

for every

n \geq 1

. Let

i

and

j

be the least numbers satisfying the conditions

a_{n+i} \equiv a_n \pmod {5}

and

a_{n+j} \equiv a_n \pmod {7}

for every

n\geq 1

. Which one below cannot be an

(i,j)

pair?

<span class='latex-bold'>(A)</span>\ (16,4) \qquad<span class='latex-bold'>(B)</span>\ (3,16) \qquad<span class='latex-bold'>(C)</span>\ (8,6) \qquad<span class='latex-bold'>(D)</span>\ (1,48) \qquad<span class='latex-bold'>(E)</span>\ (16,18)

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