Alif99 wrote:A number when divided by a divisor leaves a remainder 24. When twice the original number is divided by the same divisor the remainder is 11. What is the value of the divisor?

A)13 B)59 C)35 D)37 E)12.

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There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
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Let the divisor = d
Let the original number be N

Given: A number when divided by a divisor leaves a remainder 24
We're not told what the quotient is. So, let's just say the quotient is k
In other words: When N is divided by d, we get k with remainder 24
Applying the above rule, we can write: N = dk + 24

Also given: When twice the original number is divided by the same divisor the remainder is 11.
Once again, we're not told what the quotient is. So, let's just say the quotient here is j
In other words: When 2N is divided by d, we get j with remainder 11
Applying the above rule, we can write: 2N = dj + 11

We now have two VERY USEFUL equations:
N = dk + 24
2N = dj + 11

Take the top equation and create an EQUIVALENT equation by multiplying both sides by 2 to get:
2N = 2dk + 48
2N = dj + 11

Since both equations are set equal to 2N, we can write: dj + 11 = 2dk + 48
Subtract 11 from both sides: dj = 2dk + 37
Subtract 2dk from both sides: dj - 2dk = 37
Factor: d(j - 2k) = 37

So, d TIMES (j - 2k) = 37
Notice that d and (j - 2k) are INTEGERS.
Also, 37 is a PRIME number, which means it can be factored in only one way: 37 = (1)(37)
This means EITHER d = 1 and (j - 2k) = 37
OR d = 37 and (j - 2k) = 1

Check the answer choices....
Answer: D

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