What is the remainder?

What is the question source?

(6^35 - 6^25 )..whatever will be the value of this ..it will be a multiple of 6 or it will be divisible by 6. Example- 6^2-6 = 30 which is divisible by 6. It follows from the rule "If two multiples of a number are subtracted then the result would be another multiple of that number. Example 49-14=35..49 and 14 are multiple of 7 and the resultant is also a multiple of 7.
Now, in the question.. 6^35-6^25 will result in a multiple of 6. Subtracting 2 from a multiple of 6 will leave remainder as 4.Example- 18 is divisible by 6 and if we subtract 2 from 18 we get 16. 16÷6 will leave remainder as 4. Hence, for the question under discussion, 4 will be the remainder.

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\(6^{25}(6^{10} -1) -2 = 6k + 6 -2 = 6k +4\)
Remainder when \(6^{35} - 6^{25} - 2\) is divided by 6 is 4.

IMO D

Can you please elaborate on term "6k+6" used in your solution ? How did you get that

Recall that 6 raised to any positive integer power is always divisible by 6. Thus, we see that the first two terms are divisible by 6. Therefore, the remainder is the last term “-2.” Since -2 can’t be remainder (remainder can’t be negative), we can add 6 to -2 to obtain 4 as the actual remainder.

Answer: D

\(\frac{6^{35} - 6^{25} - 2}{6}=\frac{6^{35} - 6^{25} - 6 +4}{6}\)

\(=\frac{6(6^{34} - 6^{24} - 1) +4}{6}\)

At this point, we can see that the numerator is 4 greater than some multiple of 6,
So when we divide the numerator by 6, the remainder must be 4.

Answer: D

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