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Positive integer A > 10,000 and B > 20,000. When A is divided by 237,

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Positive integer A > 10,000 and B > 20,000. When A is divided by 237, [#permalink] New post   28 Apr 2022, 02:00
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Positive integer A > 10,000 and B > 20,000. When A is divided by 237, the remainder is 85. When B is divided by 237, the remainder is 41. What is the remainder when (2A + 3B) is divided by 237?

A. 0
B. 56
C. 123
D. 170
E. 211

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B
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Re: Positive integer A > 10,000 and B > 20,000. When A is divided by 237, [#permalink] New post   28 Apr 2022, 06:15
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Carcass wrote:
Positive integer A > 10,000 and B > 20,000. When A is divided by 237, the remainder is 85. When B is divided by 237, the remainder is 41. What is the remainder when (2A + 3B) is divided by 237?

A. 0
B. 56
C. 123
D. 170
E. 211


Key property: We can take ANY possible values of A and B (that satisfy the given information) and use them to determine the remainder when (2A + 3B) is divided by 237.
So we can ignore the information that tells us A > 10,000 and B > 20,000, and just use the smallest possible values of A and B.

Another property: If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Given: When A is divided by 237, the remainder is 85
So, 85 is the smallest possible value of A.

Given: When B is divided by 237, the remainder is 41
So, 41 is the smallest possible value of B.

Question: What is the remainder when (2A + 3B) is divided by 237?
2A + 3B = 2(85) + 3(41) = 170 + 123 = 293
293 divided by 237 equals 1 with remainder 56

Answer: B
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Re: Positive integer A > 10,000 and B > 20,000. When A is divided by 237, [#permalink] New post   08 Sep 2022, 20:09
When A is divided by 237, the remainder is 85

Theory: Dividend = Divisor*Quotient + Remainder

A -> Dividend
237 -> Divisor
a -> Quotient (Assume)
85 -> Remainders
=> A = 237*a + 85 = 237a + 85

When B is divided by 237, the remainder is 41

Let quotient be b
=> B = 237*b + 41 = 237b + 41

What is the remainder when (2A + 3B) is divided by 237

2A + 3B = 2*(237a + 85) + 3*(237b + 41) = 237*2a + 170 + 237*3b + 123 = 237*(2a + 3b) + 237 + 56 = 237*(2a + 3b + 1) + 56

=> 2A + 3B when divided by 237 gives 2a + 3b + 1 as quotient and 56 as remainder.

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Remainders

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Re: Positive integer A > 10,000 and B > 20,000. When A is divided by 237, [#permalink] New post   15 Oct 2024, 06:45
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Re: Positive integer A > 10,000 and B > 20,000. When A is divided by 237, [#permalink]
15 Oct 2024, 06:45
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