When you divide a by b, the quotient/remainder formula tells us

a = Qb + R

where Q is the quotient, R the remainder. Divide this by b on both sides and we have

a/b = Q + (R/b)

Since R < b by the definition of a remainder, on the right side above Q is the integer part of the result of the division, and R/b is the fractional or decimal part. So if a/b = 3.45, then Q = 3, and R/b = 0.45. We don't care about Q here; we only care that

R/b = 0.45
R/b = 9/20

This is a ratio of two integers, and it's fully reduced, so R must be a multiple of 9 (and b must be a multiple of 20, though we don't need that here). Since answer A is not a multiple of 9, it cannot be the value of R.

BTGmoderatorDC wrote:a and b are integers such that a/b=3.45. If R is the remainder of a/b, which of the following could NOT be equal to R?

A. 3
B. 9
C. 36
D. 81
E. 144

OA A

Source: Manhattan Prep

We can create the equation:

a/b = 3 + 45/100

a/b = 3 + 9/20

We see that the remainder is a multiple of 9, so 3 cannot be R.

Answer: A