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Question

Find the remainder when 323232 is divided by 7.


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Solution

Step 1: Expand given number

The given number is 323232=28+43232

Let us consider

323232=32k=28+4k

Using binomial expansion we get,

28+4k=C0k28k40+C1k28k-141+........+Ckk2804k

The expansion has all the factors divisible by 7 except Ckk2804k.
Step 2: Determine the cyclicity

Hence

41 when divided by 7, leaves a remainder of 4

42 when divided by 7, leaves a remainder of 2

43 when divided by 7, leaves a remainder of 1.

And then the same process of 4,2, and 1 will continue.

If the given number is of form 43k+1, a remainder of 4 is obtained.

If the given number is of form 43k+2, a remainder of 2 is obtained.

If the given number is of form 43k+1, it leaves a remainder of 1.

Step 3: Calculate the remainder

The number given is 43232

Let us find out RemPowerCyclicity.

Rem32323=Rem-1323=1

The number is of form 43k+1.

Rem432327=4

Hence, the remainder when 323232 is divided by 7 is 4.


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