Question

The least value of positive integer \(m\) for which \(361=1 \) mod(\(m\)) is Not True, is:

• A. \(m=7\)
• B. \(m=10\)
• C. \(m=11\)
• D. \(m=8\)

Answer 1

The Correct Option is A

The least positive integer \(m\) for which \(361 = 1 \mod(m)\) is not true is asked. This means we need to find the smallest \(m\) where 361 is not 1 modulo \(m\). In modulo arithmetic, \(a \equiv b \mod (m)\) shows that when \(a\) is divided by \(m\), the remainder is \(b\). To satisfy \(361 \equiv 1 \mod(m)\), the result of \(361\) divided by \(m\) should give a remainder of 1.

Let's check each option:

• For \(m=7:\) \(361 \div 7 = 51\) remainder \(4\). Thus, \(361 \equiv 4 \mod(7)\). Here, the remainder is 4, not 1, so \(m=7\) is the answer as it is not true for \(m=7\).
• For \(m=10:\) \(361 \div 10 = 36\) remainder \(1\). Thus, \(361 \equiv 1 \mod(10)\).
• For \(m=11:\) \(361 \div 11 = 32\) remainder \(9\). Thus, \(361 \equiv 9 \mod(11)\).
• For \(m=8:\) \(361 \div 8 = 45\) remainder \(1\). Thus, \(361 \equiv 1 \mod(8)\).

Therefore, the least value of \(m\) for which \(361 = 1 \mod(m)\) is not true is \(m=7\).