If c is a positive integer less than 10 and n = 9446 + c, then what is the value of c? (1) n is divisible by 5. (2) n is divisible by 9.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(B) As c is a positive integer less than 10, we are considering n to be an integer from 9447 to 9455 inclusive. Let’s consider the statement (1). For n to be divisible by 5, the last digit must be 0 or 5. So n can be 9450 or 9455. The corresponding values of c are 4 and 9. Two possible answer choices mean that we don’t have sufficient information using the statement (1) by itself. Let’s consider the statement (2). For n to be divisible by 9, the sum of digits must be divisible by 9. If the first three digits are 944, then the sum will be an integer from 9 + 4 + 4 + 7 = 24 to 9 + 4 + 4 + 9 = 26 inclusive. Obviously, none of these, 24, 25, or 26, is divisible by 9. If the first three digits of n are 945, then the sum will be an integer from 9 + 4 + 5 + 0 = 18 to 9 + 4 + 5 + 5 = 23 inclusive. Obviously, only 18 is divisible by 9, which yields 4 as the only possible value of c. Therefore the statement (2) by itself is sufficient and the correct answer is B. ---------- I was thinking of the same thing as c can be 9 but we know that 9/5 is 1 and remainder is 4 so c can never be 9.

9446 + 9 = 94505 which divisible by 9, remainder 0. So c can only be 4 unless 0 is considered as a remainder. Am I right?

First of all, be very clear on which particular statement you consider or if you consider them both. Be careful to keep in mind that an information in a statement (1) is NOT given, when you consider a statement (2) by itself and vice versa.

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I was thinking of the same thing as c can be 9 but we know that 9/5 is 1 and remainder is 4 so c can never be 9.

I assume, you are considering the statement (1) by itself. Note, that we do NOT add c as the fifth digit, but we add it to the number 9446 getting another 4-digit integer as the result (from 9447 to 9455 inclusive).

"9 / 5 is 1 and remainder is 4" is true and it would have fit the solution if the number had been 9440 + c or if c had been the fifth digit. But in our case c is NOT the last digit of n. In our case c = 9 yields n = 9446 + 9 = 9455, which IS divisible 5, because its last digit is. Therefore c can be 9, when we use the statement (1) by itself.

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9446 + 9 = 94505 which divisible by 9, remainder 0. So c can only be 4 unless 0 is considered as a remainder. Am I right?

I assume, there is a mistype and you mean "9446 + 9 = 9455". This integer is NOT divisible by 9, because 9 + 4 + 5 + 5 = 19 and 19 is NOT. In fact 9455 = 1050 × 9 + 5, so the remainder is 5.

If you mean "9446 + 4 = 9450", then you're correct about its divisibility by 9. In this case be careful when you make your decision that this is the ONLY possible value. In this particular problem we initially have only 9 possible consecutive values for c and for n as well. Therefore we quite easily can see that other 8 values do NOT fit. If the problem had been a different one, we could have been in need of more reasoning to determine that it had been the only possible value.

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