A number when divided by a divisor leaves a remainder of 24.

Question

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

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

I'm already familiar with the textbook method. I'm trying to discover what's wrong with the below method.

(1) a/d = x+24 ----> dx+24 = a
(2) 2(a)/d = x+11 ----> 2(dx+24)/d = x+11 ----> 2dx+48/d = x+11
(3) dx+11 = 2dx+48 ----> -dx=37, dx=-37

I managed to produce the correct answer. Nonetheless, there is something wrong with this. Can/will anyone help?

techjanson · Accepted Answer

See since remainder is 24 therefore the divisor D is > 24. Now by given info we have

N = (D x k) + 24
When 2N is divided by D the remainder will be 24 x 2 = 48 but since it is 11 this means that D = 48 - 11 = 37

flyingbunny · Answer

the right way to do:

n=m*x+24, and
2n=p*x+11

therefore, 2mx+48=px+11
 x=\frac{37}{p-2m}

x is an integer
therefore x=37 or x=1

seofah · Answer

I am sorry, but described steps have mistakes in it.
My quick way of solving this would be:

1) a= d+24 
2) 2a=kd+11
3) multiply step 1 by 2 and subtruct 1 from 2
4) 0=d(k-2)-37
5) because 37 is prime number, eq would make sense if k=3, hence answer would be 37 (D)

ggarr · Answer

Hi, Thanks. I know. Hence the answer. Will you let me know what mistakes you found.
Also, how does a = d+24 when the prob says "a number when divided by a divisor leaves a remainder of 24"? Is this some sort of shortcut?

andrehaui · Answer

make x/y=z+24
and 2x/y=z+11

so it equals 37

D

seofah · Answer

strictly speaking, x/y =z+24 is not correct (it implies x=zy+z24, which is not what we want to do), as garr was trying to do as well.

Garr, when I wrote a=d+24, it is of the form a=dk+r, where I really considered when k=1

vshaunak@gmail.com · Answer

Let the number is N, the divisor = D, 

I will make the two equations-
N = xD+24
2N = yD+11
where x and y are integers

Solving them: D(y-2x) = 37
as D is also integer and 37 is a prime number, the D should be 37 to satisfy the above equation.

Hence answer is 'D'

GMAT TIGER · Answer

Should be 24x2 - 11 = 37

powerka · Answer

Remainder 1 = R1 = 24
Remainder 2 = R2 = 11
R2 = 2R1 - "excess remainder" -> 11 = 48 - "excess remainder" -> "excess remainder" = 37

We know that:
. The excess remainder is a multiple of the divisor.
. The divisor is greater than the greatest remainder (which is 24).

The only multiple of 37 greater than 24 is 37. Therefore, the divisor is 37.

Lets illustrate this by picking numbers: a=24
remainder equation: a/d=k+r/d
-a. 24/37=0+24/37 ->remainder is 24
-ax2. 48/37=1+11/37 ->remainder is 11

Another way to see it is through algebra:
-a. a/d=k+24/d -> a=dk+24
-ax2. 2a/d=q+11/d -> a=(dq+11)/2
 -> 2dk+48=dq+11
 Keep in mind that dq = 2dk + "excess remainder" 
 -> 2dk + 48 = 2dk + "excess remainder" + 11
 -> "excess remainder" = 37

Sorry if this is not clear enough. Can't think of another way to explain it.

You may want to refer to the Man NP guide for looking into arithmetic with remainders (pg128).

Cheers

gmate2010 · Answer

N = I1*D + 24  --equation 1st

2N = I2*D + 11 --equation 2nd

Subtract 1st from 2nd..

=>  N = (I2 - I1)*D - 13

=>  N + 13 = I*D  ---- where I = I2-I1  wil also be an integer..

using 1st equation..

=>  I1*D + 24 +13 = I*D

=>  \frac{I1*D}{D} + \frac{37}{D} = I

=>  I1 + \frac{37}{D} = I

=>  \frac{37}{D}  should be an integer which is only possible when  D = 1 or 37

But 1 never leaves any reaminder..So,  D = 37

BrentGMATPrepNow · Answer

----ASIDE--------------
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
-------------------------

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

RELATED VIDEO 
 https://www.youtube.com/watch?v=XCxgIR042tE

CrackverbalGMAT · Answer

Let's say the divisor be K.

When a number is divided by a divisor K, leaves a remainder of 24
N= a*K + 24 
From this statement ,we can conclude that K > 24, Hence Option A,B can be eliminated

When twice the original number is divided by the same divisor, the remainder is 11.

2N = b*K + 11

2N= 2*a*K + 48

Equating both,
 b*K + 11 = 2*a*K + 48

(b-2*a)K= 48-11 = 37

Since 37 is a prime number, 1 * 37 is the only way we can write 37 as the product of two numbers.
=> (b-2*a)*K= 1* 37
from the above statement we can conclude that , Value of K = 37.

Option D is the correct answer.

Thanks,
 Clifin J Francis.
GMAT SME

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A number when divided by a divisor leaves a remainder of 24.

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