Which of the following holds true?
(A) \( 26\equiv 1\pmod 5 \)
(B) \( 17\equiv 3\pmod 8 \)
(C) \( -30\equiv -15\pmod 3 \)
(D) \( 39\equiv 12\pmod 7 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
By definition, \( a\equiv b\pmod m \) means that m divides a − b.
\( \displaystyle 26-1=25 \)
\( \displaystyle 17-3=14 \)
\( \displaystyle -30-(-15)=-15 \)
\( \displaystyle 39-12=27 \)
So the differences 25, 14, −15 and 27 must be divisible by 5, 8, 3 and 7 respectively.
Checking the statements
(A) \( 26\equiv 1\pmod 5 \) — Correct. 25 is divisible by 5.
(B) \( 17\equiv 3\pmod 8 \) — Incorrect. 14 is not divisible by 8.
(C) \( -30\equiv -15\pmod 3 \) — Correct. −15 is divisible by 3.
(D) \( 39\equiv 12\pmod 7 \) — Incorrect. 27 is not divisible by 7.