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What is the least number that when divided by 44 leaves a remainder 31

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What is the least number that when divided by 44 leaves a remainder 31 [#permalink] New post   05 Feb 2022, 08:04
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What is the least number that when divided by 44 leaves a remainder 31, when divided by 56 leaves a remainder 43, and when divided by 32 leaves a remainder 19?

A. 2464
B. 2477
C. 2451
D. 616
E. 603
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Re: What is the least number that when divided by 44 leaves a remainder 31 [#permalink] New post   05 Feb 2022, 08:37
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There are more systematic ways to do questions like this, but we don't need to bother with those, because we know one of the answer choices is correct. We get an odd remainder dividing by 32, so our number must be odd, leaving B, C or E as the answer. We can test 603: since 640 - 64 = 576 is a multiple of 32, the remainder when we divide 603 by 32 is not 19 (it is 603 - 576 = 27), so answer E is wrong. The remaining two answers are 26 apart, so they'll definitely give different remainders when we divide by any of the three numbers in the question (since those numbers are all larger than 26), so we now can just divide one of the two remaining answer choices by one of 32, 44 or 56, and find the remainder, and if the remainder is right, the answer is right. Or we can locate a multiple of, say, 44, that's nearby: since 44*50 = 2200, then 44*55 = 2200 + 220 = 2420, and when we divide 2451 by 44, we see we get a remainder of 31, so 2451 must be the answer.
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What is the least number that when divided by 44 leaves a remainder 31 [#permalink] New post   Updated on: 17 Feb 2022, 05:12
Looking back at my suggested solution for the exercise it was not correct, so I had to remove it. Sorry for the confusion.
(for those who read it... something that is divisible by a smaller factor of the numbers given e.g. divisible by 8 is not necessarily also divisible by 32)

Originally posted by Markella24 on 16 Feb 2022, 14:18.
Last edited by Markella24 on 17 Feb 2022, 05:12, edited 3 times in total.
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Re: What is the least number that when divided by 44 leaves a remainder 31 [#permalink] New post   17 Feb 2022, 04:41
Markella24 wrote:
We notice here that all 44, 56 and 32 are multiples of 4. So, to help our calculations we can rephrase as follows:
What is the least number that when divided by 11 leaves a remainder 9, when divided by 14 leaves a remainder 11, and when divided by 8 leaves a remainder 3? (notice here that I calculated new remainders based on the new divisors so e.g. 11 divides 31 and leaves a remainder of 9).

Next, I am attacking the answer choices starting from choice C (middle value). In this case, it does not really matter where you start but it is usually helpful to start in the middle.

I choose to test with 11.
So, I take my answer choice and substract the remainder i.e. 2451-9=2442
According to the divisibility rules if a number is a multiple of 11 then by substracting the value of the last digit from the rest of the number you should get a multiple of 11 and this is a repeatable process.

Therefore,
244-2=242, which is a multiple of 11! However, if you were not able to identify that, you could repeat the process and get:
24-2=22, here you should easily identify that this is a multiple of 11. And there you have your answer C.


Shouldn't the remainder be 1 when the number is divided by14? 43 = 14*3 + 1.
or am i missing something here?

and good take on divisibility by 11. This method seems simpler than the difference of odd and even values rule, provided i can remember it :)
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Re: What is the least number that when divided by 44 leaves a remainder 31 [#permalink] New post   09 Apr 2022, 03:58
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Divided by 44 leaves a remainder of 31 therefore when divided by 11 number will leave a remainder of 9. only c and e satisfy this criterion.
Divided by 56 leaves a remainder of 43 therefore when divided by 7 number will leave a remainder of 1. both c and e satisfy this criterion.
Divided by 32leaves a remainder of 19 therefore when divided by 16 number will leave a remainder of 3. only c satisfies this criterion.
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Re: What is the least number that when divided by 44 leaves a remainder 31 [#permalink] New post   10 Mar 2024, 00:19
There is another way of solving such questions, the more "certain" way and it takes very much the same amount of time as answer choice elevation.
Find out the LCM of the divisors 32, 44 & 56. This value comes out as 2464 and this is the lowest number that can divide each of 32, 44 & 56 and leave no remainder.
So, our answer is 2464 - (divisor) + (remainder). Put in any divisor and its corresponding remainder, and you have your answer.
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Re: What is the least number that when divided by 44 leaves a remainder 31 [#permalink]
10 Mar 2024, 00:19
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