# CAT Practice : Number System - Remainders

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When we add two numbers that are not coprime, the sum of these two numbers cannot be prime. Awesome intuitive stuff, but very often forgotten. Also, learn something from Euler today.

## Remainders, coprime numbers

Q.7: A prime number p greater than 100 leaves a remainder q on division by 28. How many values can q take?
1. 8
2. 12
3. 9
4. 15

Choice B. 12

## Detailed Solution

q can be 1.
If q =2, number would be of the form 28n + 2 which is a multiple of 2.

Similarly, when q = 4, number would be of the form 28n + 4 which is again a multiple of 2. Any number of the form 28n + an even number will be a multiple of 2.

When q = 7, number would be of the form 28n + 7 which is a multiple of 7.

So, the only remainders possible are remainders that share no factors with 28. Or numbers that are co-prime to 28.

There is a formula for this and a shorter way of finding the number of numbers co-prime to a given natural number. A more detailed discussion on this is provided here and here.

1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25 and 27. q can take 12 different values.

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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.