We shall use the following well-known properties of congruences:
1) if a≡b(modm), then a≡b+m⋅k(modm);
2) if ad≡bd(modm) and gcd(d,m)=1, then a≡b(modm).
Let us solve the linear congruence 34x≡53(mod89). This congruence is equivalent to 34x≡(53+89)(mod89), that is 34x≡142(mod89). Since gcd(2,89)=1, we conclude that after dividing both parts by 2 the last congruence is equivalent to 17x≡71(mod89). Then we have 17x≡(71−5⋅89)(mod89), and hence 17x≡−374(mod89). Taking into account that gcd(17,89)=1, we conclude that that after dividing both parts by 17 the solution of the congruence is x≡−22(mod89)≡67(mod89).
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