If 1350 is divided by 14, the remainder is

When a number is exactly one less than a multiple of the divisor — that is, it leaves a residue of -1 when divided by the divisor — raising it to a power reduces to raising -1 to that same power: the result is 1 when the exponent is even, and the divisor’s largest residue (one less than the divisor) when the exponent is odd, because a negative sign multiplied an even number of times cancels out, while an odd number of multiplications leaves it in place.

1. Write 13 in terms of 14: 13 = 14 − 1, so 13 leaves a residue of -1 when divided by 14.

2. Raise both sides to the 50th power: 1350 has the same residue, modulo 14, as (-1)50.

3. Check the parity of the exponent: 50 is an even number.

4. Because the exponent is even, (-1)50 equals 1, so 1350 leaves a residue of 1 when divided by 14.

Independent check with a smaller power first: 132 = 169, and 169 = 14 × 12 + 1, so 132 already leaves a residue of 1 when divided by 14. Since 50 = 2 × 25, 1350 = (132)25, whose residue is 125 = 1 — the same result reached through the parity argument above, so the remainder when 1350 is divided by 14 is 1.