Find the remainder when 961¹²⁵ divided by 37.

Given:

Expression: \(961^{125}\) divided by 37.

Formula used:

Modular arithmetic property: If \(a \equiv b \pmod{m}\), then \(a^n \equiv b^n \pmod{m}\).

Also, a negative remainder \(-k \pmod{m}\) is equivalent to \((m - k) \pmod{m}\).

Calculations:

Divide 961 by 37:

961 = 37 × 25 + 36

So, 961 leaves a remainder of 36 when divided by 37.

This can be written as: \(961 \equiv 36 \pmod{37}\)

Since 36 is one less than 37, we can write:

\(36 \equiv -1 \pmod{37}\)

\(961^{125} \equiv (-1)^{125} \pmod{37}\)

Since 125 is an odd number, \((-1)^{125} = -1\).

So, \(961^{125} \equiv -1 \pmod{37}\)

To get a positive remainder, add the modulus (37) to the negative remainder:

\(-1 \equiv -1 + 37 \pmod{37}\)

\(-1 \equiv 36 \pmod{37}\)

∴ The remainder when \(961^{125}\) is divided by 37 is 36.