Remainder

[ela07mjt]

When the positive integer n is divided by 25, the remainder is 13. What is the value of n ?

(1) n < 100
(2) When n is divided by 20, the remainder is 3.

[hemant_rajput]

When the positive integer n is divided by 25, the remainder is 13. What is the value of n ?

(1) n < 100
(2) When n is divided by 20, the remainder is 3.

n will be of form 25k -13

statement 1:-

k is between 1 to 4
because 25*4 = 100 and n should be less than 100.

so now there are 4 possible values of n - 12,37,62,87
NOT Sufficient.

statement 2 :-

n is also of for 20J - 3

20J -3 = 25K -13
25K - 20J = 10 =? 5K - 4J = 2

for K = 1 , J will 3/4, not possible has to be +ve integer.
K =2 , J = 2. Possible value of J
K = 3, J = 13/4, not possible has to be +ve integer.
K = 4, J = 17/4, not possible has to be +ve integer.
K = 5, J = 22/4, not possible has to be +ve integer.
k = 6, J = 7 and so on.

so not sufficient.

Now combining two statement only one value exist which satisfy both equation
K =2 , J = 2, i.e. 37

Answer C

[Anurag@Gurome]

When the positive integer n is divided by 25, the remainder is 13. What is the value of n ?

(1) n < 100
(2) When n is divided by 20, the remainder is 3.

n will be of the form (25m + 13), where m is some non-negative integer.
Hence, possible values of n are 13, 38, 63, 88, 113, 138, 163 ... etc

Statement 1: n can be 13 or 38 or 63 or 88 ---> Not sufficient

Statement 2: n is also of the form (20n + 3), where n is some non-negative integer.
Hence, n can be 63, 163 ... etc.

Not sufficient

1 & 2 Together: n can only be 63 ---> Sufficient

The correct answer is C.

[Anurag@Gurome]

n will be of form 25k -13
...
so now there are 4 possible values of n - 12,37,62,87
NOT Sufficient.

Hi Hemant!

Although you have achieved the "correct answer" that too using the proper algebraic method, I'd chip in to point out a mistake which may be a silly mistake or a basic misconception.

If a positive integer n leaves a remainder of 13 when divided by 25, n will either of the form (25k + 13) or (25j - 12), where k is some non-negative integer and j is some positive integer.

You could've clearly checked this with the possible values of n you've got.
12, 37, 62, and 87 - all of them leaves a remainder of 12 (not 13) when divided by 25.

Hope this helps.

[hemant_rajput]

n will be of form 25k -13
...
so now there are 4 possible values of n - 12,37,62,87
NOT Sufficient.

Hi Hemant!

Although you have achieved the "correct answer" that too using the proper algebraic method, I'd chip in to point out a mistake which may be a silly mistake or a basic misconception.

If a positive integer n leaves a remainder of 13 when divided by 25, n will either of the form (25k + 13) or (25j - 12), where k is some non-negative integer and j is some positive integer.

You could've clearly checked this with the possible values of n you've got.
12, 37, 62, and 87 - all of them leaves a remainder of 12 (not 13) when divided by 25.

Hope this helps.

Hi Anurag,


Thanks for pointing it out. I jumbled the concepts.lol

Cheers,
Hemant

[GMATGuruNY]

A quick lesson on remainders:

When x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3.

When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4.

Combined, the statements above imply that when x is divided by both 5 and 7 -- in other words, when x is divided by 35 -- there will be a constant remainder R.
Put another way, x is R more than a multiple of 35:
x = 35c + R.

To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13, 18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11, 18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.

Putting it all together:
x = 35c + 18.

Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10, 13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2, 13...

Thus, when x is divided by both 3 and 11 -- in other words, when x is divided by 33 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...

Onto the problem at hand:

When the positive integer n is divided by 25, the remainder is 13. What is the value of n ?

(1) n < 100
(2) When n is divided by 20, the remainder is 3.

According to the question stem:
n = 25a + 13 = 13, 38, 63, 88...

Statement 1: n<100
n could be any of the values in the list above.
INSUFFICIENT.

Statement 2: When n is divided by 20, the remainder is 3.
Thus:
n = 20b + 3 = 3, 23, 43, 63...
When we combine this condition with that in the question stem, we know the following:
n must be a multiple of 20 and 25 -- in other words, a multiple of 100 -- plus the smallest value common to both lists (63).
Thus:
n = 100c + 63 = 63, 163, 263...
Since n can be more than one value, INSUFFICIENT.

Statements combined:
The only value that satisfies both statements is n=63.
SUFFICIENT.

The correct answer is C.