Question #267936

Solve the linear congruence 34x ≡ 53(mod 89).


1
Expert's answer
2021-11-21T16:46:19-0500

We shall use the following well-known properties of congruences:

1) if ab(modm),a≡b (\mod m), then ab+mk(modm);a≡b+m\cdot k (\mod m);

2) if adbd(modm)ad≡bd (\mod m) and gcd(d,m)=1,gcd(d,m)=1, then ab(modm).a≡b (\mod m).


Let us solve the linear congruence 34x53(mod89).34x ≡ 53(\mod 89). This congruence is equivalent to 34x(53+89)(mod89),34x ≡ (53+89)(\mod 89), that is 34x142(mod89).34x ≡ 142(\mod 89). Since gcd(2,89)=1,gcd(2,89)=1, we conclude that after dividing both parts by 2 the last congruence is equivalent to 17x71(mod89).17x ≡ 71(\mod 89). Then we have 17x(71589)(mod89),17x ≡ (71-5\cdot 89)(\mod 89), and hence 17x374(mod89).17x ≡ -374(\mod 89). Taking into account that gcd(17,89)=1,gcd(17,89)=1, we conclude that that after dividing both parts by 1717 the solution of the congruence is x22(mod89)67(mod89).x ≡ -22(\mod 89)≡67(\mod 89).

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