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

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Math Expert
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What is the remainder when the positive integer x is divided by 31? [#permalink]

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14 Aug 2017, 03:51
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35% (medium)

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73% (01:18) correct 27% (01:40) wrong based on 96 sessions

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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.

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Re: What is the remainder when the positive integer x is divided by 31? [#permalink]

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14 Aug 2017, 04:12
2
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.
The number x can take the value 21,52,83.... and so on seperated by 31.
The remainder of the numbers x when divided by 31 will always be 21(Sufficient)

2. When x + 33 is divided by 10 the remainder is zero.
The number x=7, gives a remainder of 0 when divided by 10.
7 divided by 31 gives a remainder of 7
If x=37, the remainder of x + 33 is 0 when divided by 10.
But, 37 divided by 31 will have a remainder of 6.(Insufficient)(Option A)
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Re: What is the remainder when the positive integer x is divided by 31? [#permalink]

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14 Aug 2017, 04:39
pushpitkc 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.
The number x can take the value 21,52,83.... and so on seperated by 31.
The remainder of the numbers x when divided by 31 will always be 21(Sufficient)

2. When x + 33 is divided by 10 the remainder is zero.
The number x=7, gives a remainder of 0 when divided by 10.
7 divided by 31 gives a remainder of 7
If x=37, the remainder of x + 33 is 0 when divided by 10.
But, 37 divided by 31 will have a remainder of 6.(Insufficient)(Option A)

but if I divide 21 by 31 the remainder is not 21 ?:)
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Re: What is the remainder when the positive integer x is divided by 31? [#permalink]

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14 Aug 2017, 05:04
dave13 wrote:
pushpitkc 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.
The number x can take the value 21,52,83.... and so on seperated by 31.
The remainder of the numbers x when divided by 31 will always be 21(Sufficient)

2. When x + 33 is divided by 10 the remainder is zero.
The number x=7, gives a remainder of 0 when divided by 10.
7 divided by 31 gives a remainder of 7
If x=37, the remainder of x + 33 is 0 when divided by 10.
But, 37 divided by 31 will have a remainder of 6.(Insufficient)(Option A)

but if I divide 21 by 31 the remainder is not 21 ?:)

Hi dave13,

If the number is 21 and is divided by 31, the quotient is 0 and remainder is 21.
Hope it is clear!
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Re: What is the remainder when the positive integer x is divided by 31? [#permalink]

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14 Aug 2017, 06:16
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.

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.

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Re: What is the remainder when the positive integer x is divided by 31? [#permalink]

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14 Aug 2017, 07:10
which approach is more fool proof? number picking can be a bit difficult with more complicated questions?
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Re: What is the remainder when the positive integer x is divided by 31? [#permalink]

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24 Aug 2017, 13:37
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.

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Re: What is the remainder when the positive integer x is divided by 31?   [#permalink] 24 Aug 2017, 13:37
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