N is a positive integer
quantity A - Remainder when 2^N is divided by 10
quantity B - remainder when 7^N is divided by 5
Write down 7 as 2+5
Now, in the binomial expansion of (2+5)^N, you get 2^N + 5^N + (other terms which have both 5 and 2)
Take for example (2+5)^2 = 2^2 + 5^2 + 2.5.2
Now, in the expansion, you can see that except for the first term, all other terms are perfectly divisible by 5 (because they contain at least one factor of 5), so the only reminder you’ll get will be because of 2^N/5, so that simplifies the option B, now you can easily compare.
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sir , I was unable to understand. is there any other way to solve this .
You can try by choosing numbers
N = 1,2,3,4,5,6…
2^N = 2,4,8,16,32,64
Quantity A = 2,4,8,6,2,4…
7^N = 7,49,343,2401,16807,117649,…
Quantity B = 2,4,3,1,2,4…
Compare A and B
2 = 2
4 = 4
8 > 3
6 > 1
2 = 2
4 = 4
…
…
Answer would be D
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