The least value of positive integer m for which 361 = 1 mod(m) is Not True, is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → m = 7
Here “361 = 1 mod(m)” means the congruence \(361 \equiv 1 \pmod{m}\). It is true exactly when m divides the difference:
\(361 - 1 = 360 = 2^3 \times 3^2 \times 5\)
So the congruence is not true when m is not a divisor of 360. Checking m = 1, 2, 3, … in turn:
1, 2, 3, 4, 5 and 6 all divide 360.
7 does not divide 360 (\(360 = 7 \times 51 + 3\)).
So the least such m is 7.
Why the other options are incorrect
Option (2) m = 10 — Incorrect. 10 divides 360, so the congruence is true for m = 10.
Option (3) m = 11 — Incorrect. The congruence is not true for m = 11, but 7 is a smaller such value.
Option (4) m = 8 — Incorrect. 8 divides 360 (360 = 8 × 45), so the congruence is true for m = 8.