Data Sufficiency

If j and k are positive integers, what is the remainder when 8 × 10^k + j is divided by 9?

[1] k = 13.

[2] j = 1.



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Sanju09!

IMO B

Sum of digits of the number 8*10^k + j is equal to 8 + j.
The number 8*10^k + j is divisible by 9 without remainder if 8 + j = 9
Remainder is equal to 8 + j - 9 = j - 1

(1) k = 13
Statement (1) ALONE is INSUFFICIENT.
We should eliminate answer choices A and D.

(2) j = 1
Remainder is equal to 0.
Statement (2) ALONE is SUFFICIENT.
We should eliminate answer choices C and E.
I choose B.

Hope it helps!
Best,
Maciek

Great work, Maciek - really nice use of the divisibility-by-9 rule!

A few other thoughts on this one:

1) It should seem obvious that you could solve the problem with BOTH statements...which is why it's highly unlikely that the answer would be an easy C. If you can look at a DS question and without much thought determine that both statements together are sufficient, you're probably looking at answer A or B, which is why you should:

2) Try to see which piece of information you don't need. Statement 1 to me immediately seemed a little unnecessary, as I know that exponents tend to be very pattern-driven, particularly base 10 exponents.

As a result, I just set out to find a quick pattern that would let me know that 8*10^k would produce the same remainder each time:

80 ---> 72 is a multiple of 9, so the remainder is 8
800 --> 720 is a multiple of 9 and so is 72, so that gives you 792 and a remainder of 8
8000 --> 7920 we know from above is a multiple of 9 and so is 72, so 7992 is a multiple of 9 and you have a remainder of 8

Quickly you can see the pattern - keep adding multiples of 72 and you'll always end up with the same remainder (8) for 8*10^k, so it doesn't matter what k is. What really matters is j.

When you then see that statement 2 gives you j, you know that you'll have sufficient information with 2 but not 1, so the correct answer is B.



This problem brings up a few high-level strategic points:

1) If C seems too easy, it probably is - see which piece of information you don't need.

2) Exponents are incredibly pattern-driven, so for divisibility with exponents see if you can establish a pattern that allows you to work with smaller numbers to prove a point about the entire range of exponents.