Bunuel wrote:

What is the remainder when the positive integer x is divided by 31?

(1) When x + 10 is divided by 31 the remainder is zero.

(2) When x + 33 is divided by 10 the remainder is zero.

We use the “quotient remainder theorem,” which states: dividend = quotient x divisor + remainder.

We need to determine the remainder when the positive integer x is divided by 31. That is, if we express x as 31Q + R for some nonnegative integers Q and R where R < 31, then R is the remainder when x is divided by 31.

Statement One Alone:

When x + 10 is divided by 31, the remainder is zero.

Thus:

(x + 10)/31 = Q + 0/31

(x + 10)/31 = Q

x + 10 = 31Q

x = 31Q - 10

x = 31Q - 31 + 31 - 10

x = 31(Q - 1) + 21

Since Q - 1 is an integer and since 21 < 31, we see that when x is divided by 31, the remainder is 21.

Statement one alone is sufficient to answer the question.

Statement Two Alone:

When x + 33 is divided by 10, the remainder is zero.

Thus:

(x + 33)/10 = Q + 0/31

(x + 33)/10 = Q

x + 33 = 10Q

x = 10Q - 33

We see that if Q = 4, then x = 7 and the remainder when x is divided by 31 is 7. However, if Q = 5, then x = 17 and the remainder when x is divided by 31 is 17. Statement two is not sufficient to answer the question.

Answer: A

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