When a certain number N is divided by d, the remainder is 7. If the original number N is multiplied by 5 and then divided by d, the remainder is 10.
Find d.
The answer is d=25; N is any of 7, 32, 57, 82, 107, 132 etc etc. It was set up in a spreadsheet and arrived at by trial and error, so just knowing the answer probably won't get many marks. I'd be interested to know if anyone else can produce a formula for deriving the solution algebraically - because I can't see how to do it either.
To find the value of 'd', we need to set up a system of equations based on the given information.
Let's assume the original number 'N' can be written as:
N = qd + 7 ----(Equation 1)
where 'q' represents the quotient.
According to the second condition, when N is multiplied by 5 and divided by 'd', the remainder is 10.
This can be written as:
5N/d = sd + 10 ----(Equation 2)
where 's' represents the updated quotient.
To solve for 'd', we can eliminate 'N' by equating the right-hand sides of Equation 1 and Equation 2.
qd + 7 = sd + 10
Rearranging the terms in the equation, we get:
(q - s)d = 3
Since 'd' is a positive integer and 3 is a prime number, the only possibility is that (q - s) = 3 and d = 1.
Thus, the value of 'd' is 1.
Note: In this case, there is no unique solution for 'd' as any integer value can satisfy the conditions mentioned in the problem. The given information allows for multiple values of 'd' to exist.