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These three integers cannot all be prime

Source: South African MO 2004 Q6

May 27, 2012

inductionnumber theory unsolvednumber theory

Problem Statement

The numbers

a_1,a_2

and

a_3

are distinct positive integers, such that
(i)

a_1

is a divisor of

a_2+a_3+a_2a_3

;
(ii)

a_2

is a divisor of

a_3+a_1+a_3a_1

;
(iii)

a_3

is a divisor of

a_1+a_2+a_1a_2

.
Prove that

a_1,a_2

and

a_3

cannot all be prime.

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