Find the values of A, B, C, and D
given the following information:
1. A, B, C, and D are all unique
positive integers between 1 and 9 (no two variables have the same value).
2. A3 + B3 = C3 - D3
3. C = B + D
ANSWER: There are two solutions:
A = 6, B = 8, C = 9, D = 1 and A = 6, B = 1, C = 9, D = 8
EXPLANATION: Replacing
C with (
B + D) yields:
A
3 + B
3 = (B + D)
3 - D
3
A
3 + B
3 = (B + D)(B + D)(B + D) - D
3
This ultimately reduces to A
3 = 3BD(B + D). Since
C = B + D,
this can be rewritten as
A3 = 3BCD. So the
cube of
A is divisible by 3. The only possible values of
A are 3, 6,
or 9 (none of the other single digit integers cubed are divisible by 3). If
A was 3, the
equation reduces to 9 = BCD and there aren't three
different integers between 1 and 9 that when
multiplied together have a product of 9 so we know that
A is NOT 3. If
A
was 9, the equation reduces to 243 = BCD and there aren't three
different integers between 1 and 9
that when multiplied together have a product of 243 so we know that
A is NOT 9.
A must be 6 therefore and the equation reduces to 72 = BCD. The only three
different integers between 1 and 9 that when multiplied together make 72 are 1, 8, and 9 OR 2, 4
and 9. However, only 1, 8, and 9 can satisfy the equation
C =
B +
D (i.e. where the highest number is the sum of the other two).
C is the
largest of the three, so
C = 9. Either
B = 8 and
D = 1
or
B = 1 and
D = 8.
