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

Hi dave13,

If the number is 21 and is divided by 31, the quotient is 0 and remainder is 21.
Hope it is clear!

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

Now, we can solve this question algebraically or with number plugging, which in this case is easier:

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

Number picking approach:
x+10 to be multiple of 31 x must be 21, 52, 83, ... each yields a remainder of 21 upon division by 31. Sufficient.

Algebraic approach:
\(x+10=31q\) --> \(x=31q-10\) --> \(x=31q-31+21\) --> \(x=31(q-1)+21\) directly tells us that remainder upon division \(x\) by 31 is 21. Sufficient.

(2) When x+33 is divided by 10 the remainder is zero --> x+33 to be multiple of 10 x must be 7, 17, 27, ... 7 yields a remainder of 7 upon division by 31, but 17 yields a remainder of 17 upon division by 31. Not sufficient.

Answer: A.

which approach is more fool proof? number picking can be a bit difficult with more complicated questions?

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