Find the remainder when 247 is divided by 7.

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answer is C.

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Detailed Solution

Here we have to use the binomial theorem so as to find the remainder when 247 is divided by 7 .
The Binomial Theorem states that, where n is a positive integer:

{(a + b)}^{n} = \sum_{k = 0}^{n} (n k) a^{n - k} b^{k} = \sum_{k = 0}^{n} (n k) a^{k} b^{n - k}

⇒(a+b)n=an+nC1an−1b+nC2an−2b2+...+nCn−1abn−1+bn
So, here let a+b=2 and n=47 and we assume it to be divisible by 7,
(2)47=x(mod7)
where x is the remainder on dividing the L.H.S by 7. The notation (mod 7) denotes that the number is being divided by 7.
(22)(245)=x(mod7)
⇒4.(23×15)=x(mod7)
⇒4.(23)15=x(mod7)
⇒4.(8)15=x(mod7)
⇒4.(7+1)15=x(mod7)
Here, we have written powers of 2 in such a way that we have used the least integer that can be expressed as a sum of 7 with another integer i.e. 8=7+1. Why this is done will be understood in further steps. Now, we are expanding (7+1)15 using the binomial theorem.
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Now, on the L.H.S, we can see that from the first to the second last term we can take 7 as common and let the remaining term in the bracket be λ

⇒7λ+4=x(mod7)
⇒x=4
Now because the first term is a multiple of 7, therefore the remainder will be 0, so the extra 4 would be left as remainder on dividing the whole term by 7.
Therefore, the correct answer on dividing 247 by 7 is 4
So, the correct answer is “4”.

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