Integers

The least positive integer

m

for which

361%20%3D%201%20%5Cmod(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%20%5Cequiv%20b%20%5Cmod%20(m)

shows that when

a

is divided by

m

, the remainder is

b

. To satisfy

361%20%5Cequiv%201%20%5Cmod(m)

, the result of

361

divided by

m

should give a remainder of 1.

Let's check each option:

For $m%3D7%3A$ $361%20%5Cdiv%207%20%3D%2051$ remainder $4$ . Thus, $361%20%5Cequiv%204%20%5Cmod(7)$ . Here, the remainder is 4, not 1, so $m%3D7$ is the answer as it is not true for $m%3D7$ .
For $m%3D10%3A$ $361%20%5Cdiv%2010%20%3D%2036$ remainder $1$ . Thus, $361%20%5Cequiv%201%20%5Cmod(10)$ .
For $m%3D11%3A$ $361%20%5Cdiv%2011%20%3D%2032$ remainder $9$ . Thus, $361%20%5Cequiv%209%20%5Cmod(11)$ .
For $m%3D8%3A$ $361%20%5Cdiv%208%20%3D%2045$ remainder $1$ . Thus, $361%20%5Cequiv%201%20%5Cmod(8)$ .

Therefore, the least value of

m

for which

361%20%3D%201%20%5Cmod(m)

is not true is

m%3D7

Chapter Questions
1 MCQs