Finding Remainders: Integer Division by 7 and 4

We can start by using the remainder theorem, which states that when an integer n is divided by a positive integer d, the remainder r will be an integer between 0 and d-1.

Using this theorem, we can write:

n = 7q + 3 (where q is the quotient when n is divided by 7) n = 4p + R (where p is the quotient when n is divided by 4)

We want to find the possible values of R, so we can eliminate q by substituting the first equation into the second:

7q + 3 = 4p + R

Simplifying this equation, we get:

R = 4p - 7q + 3

This tells us that R will be an integer if 4p - 7q is 1 or 2. We can see this by testing some values of p and q:

• If p = 1 and q = 1, then R = 4(1) - 7(1) + 3 = 0
• If p = 2 and q = 3, then R = 4(2) - 7(3) + 3 = 1
• If p = 3 and q = 5, then R = 4(3) - 7(5) + 3 = 2
• If p = 4 and q = 7, then R = 4(4) - 7(7) + 3 = -1 (not an integer)

Therefore, the possible values of R are 0, 1, or 2.