Number Theory : Remainders of division

Finding remainders when the same power of two numbers leave the same remainder when divided by a common integer.

Question

For what value of 'n' will the remainder of 351ⁿ and 352ⁿ be the same when divided by 7?

1. 2
2. 3
3. 6
4. 4
   

Correct Choice is (2) and the Correct Answer is 3

Explanatory Answer

When 351 is divided by 7, the remainder is 1.

When 352 is divided by 7, the remainder is 2.

Let us look at answer choice (1), n = 2

When 351² is divided by 7, the remainder will be 1² = 1.

When 352² is divided by 7, the remainder will be 2² = 4.

So when n = 2, the remainders are different.

When n = 3,

When 351³ is divided by 7, the remainder will be 1³ = 1.

When 352³ is divided by 7, the remainder will be 2³ = 8.

As 8 is greater than 7, divide 8 again by 7, the new remainder is 1.

So when n = 3, both 351ⁿ and 352ⁿ will have the same remainder when divided by 7.

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