The question is from CAT Number Theory - Remainders. We need to find the number of possible values of remainders. A range of CAT questions can be asked based on the concept of Remainders. Another questions in the genre of possible remainders.

Question 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?

- 3
- 2
- 5
- 4

5

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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

The question is **"How many possible values can k take?"**

Choice C is the correct answer.

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