Question

Calculate the modulo operations given below: 7⁵ mod 15 =

Answer 1

Answer: 7

Work Shown

7^2 = 49

Divide this over 15 and we get 49/15 = 3 remainder 4

This would mean
`7^2 \equiv 4 \ (\text{mod 15})`

The modulo cares only about the remainder.

Square both sides.

`7^2 \equiv 4 \ (\text{mod 15})\\\\(7^2)^2 \equiv 4^2 \ (\text{mod 15})\\\\7^4 \equiv 16 \ (\text{mod 15})\\\\7^4 \equiv 1 \ (\text{mod 15})\\\\`

The last step is to multiply both sides by 7.

`7^4 \equiv 1 \ (\text{mod 15})\\\\7*7^4 \equiv 7*1 \ (\text{mod 15})\\\\7^5 \equiv 7 \ (\text{mod 15})\\\\`

Therefore, the calculation (7^5)/15 results in some quotient and a remainder of 7.

Answer 2

7

Explanation:

To calculate 7⁵ mod 15, we need to find the remainder when 7⁵ is divided by 15.

Let's break down the steps to simplify the calculation:

Step 1: Calculate 7⁵

7⁵ = 7 × 7 × 7 × 7 × 7 = 16807

Step 2: Divide 16807 by 15

16807 ÷ 15 = 1120 remainder 7

Step 3: The remainder is the result of the modulo operation

So, 7⁵ mod 15 = 7

Therefore, when we calculate 7⁵ mod 15, the answer is 7.