Question

Check whether the following congruence are true and false

(1) 9 ≡ 23 (mod 12)

(2) 15 ≡ 29 (mod 7)

(3) 175 ≡ 5 (mod 17)

(4) 21 ≡ − 8 (mod 6)

Answer 1

⇒ m divides (a − b) i.e., (a − b) is a multiple of m

(1)

Here, 9 – 23 = – 14

–14 is not a multiple of 12.

Therefore, the given congruence is false.

(2)

Here, 15 – 29 = –14

–14 is a multiple of 7.

Therefore, the given congruence is true.

(3) .

Here, 175 – 5 = 170

170 is a multiple of 17.

Therefore, the given congruence is true.

(4) .

Here, 21 – (–8) = 21 + 8 = 29

29 is not a multiple of 6.

Therefore, the given congruence is false.