A number leaves a remainder 3 on division by 14 and leaves a remainder k on division by 35 How many ...........

Explanation:

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)
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