When 43 is divided by x, the remainder is x - 5. If x is a natural number, how many solutions will x have?

Question: When 43 is divided by x, the remainder is x - 5. If x is a natural number, how many solutions will x have?

Solution:
Given that,
43 is divided by x, remainder is x - 5
And x is a natural number.

Now,
43 = kx + (x - 5) where k is an integer.
⇒ 43 = kx + x - 5
⇒ 43 = x(k + 1) - 5
⇒ x(k + 1) = 48
∴ x is a factor of 48.
Remainder r = x - 5 must be ≥ 0
⇒ x - 5 ≥ 0
⇒ x ≥ 5
Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Since x is a natural number and x - 5 must be a remainder,
Valid factors of 48 greater than 5 are 6, 8, 12, 16, 24, 48.

There are 6 factors.