# CAT Practice : Number System - Remainders

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Another in the genre of possible remainders.

## Remainders

Q.10: A number leaves a remainder 3 on division by 14, and leaves a remainder k on division by 35. How many possible values can k take?
1. 3
2. 2
3. 5
4. 4

Choice C. 5

## Detailed Solution

Let us have a look at the theory for this question as well. For instance, let us assume a number N leaves a remainder of 3 on division by 8. What would be the remainder when number N is divided by 24?

N / 8
remainder = 3
N / 24
remainder = ?
Let us look at Numbers that leave remainder 3 on division by 8
3, 11, 19, 27, 35, 43 ……

For these numbers, remainders when divided by 24 are
3, 11, 19, 3, 11, 19 ……

Possible remainders are 3, 11 or 19

Alternative Method:

N / 8
remainder = 3
N = 8q + 3

q can be in one of 3 forms
3p
3p + 1
3p + 2

N = 8(3p) + 3 or
8(3p + 1) + 3 or
8(3p + 2) + 3
24p + 3 or
24p + 11 or
24p + 19

N / 24
possible remainders are 3, 11, 19

Why did we choose to write q as 3p, 3p + 1 or 3p + 2?
8 x 3 = 24, this is why we chose 3p, 3p + 1, 3p + 2

So, if we are given that remainder on dividing N by 8, then there will be a set of possibilities for the remainder of division of N by 24 (or any multiple of 8)

Let us look at the opposite also. Say, we know the remainder of division of N by 42 is 11, what should be the remainder when N is divided by 7?
N / 42
remainder = 11
N / 7
remainder = ?

N / 42
remainder = 11
N = 42q + 11

42q + 11 divided by 7
42q leaves no remainder
11 / 7
remainder = 4

So, if we are given that remainder on dividing N by 42, then we can find the remainder of dividing N by 7 (or any factor of 42)

Now, let us address the question

A number leaves a remainder of 3 on division by 14, or it can be written as 14n + 3
On division by 70, the possible remainders can be 3, 17 (3 + 14), 31 (3 + 28), 45 (3 + 42), or 59 (3 + 56). The number can be of the form

70n + 3
70n + 17
70n + 31
70n + 45
70n + 59

Now, we need to divide this number by 35
70n + 3 divided by 35, the remainder will be 3.
70n + 17 divided by 35, the remainder will be 17.
70n + 31 divided by 35, the remainder will be 31.
70n + 45 divided by 35, the remainder will be 10.
70n + 59 divided by 35, the remainder will be 24.

On division by 35, the possible remainders are 3, 17, 31, 10 or 24. There are 5 possible remainders

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## More questions from Number Theory - Remainders

When we divide 24 pigeons into 5 groups, 4 are left out or remain. This is the remainder. When there is a question based on this idea in the exam, we have a CAT among the pigeons.