Official Explanation from MagooshAAB × B = CB5B
The first thing we should note is the role of B—it's the units digit of both factors and of the product. So, it must be a number that we can square and result in a units digit that equals the number that we squared. That's a mouthful. Let's see some examples.
0 * 0 = 0
1 * 1 =
1This is a very common trick on variables-for-digits questions like this, and it's good to know (without doing the math) that the four possible digits that result in a number with the same units digit when squared are 0, 1, 5, and 6.
If you didn't know that immediately, then check:
2 * 2 = 4 [no good]
3 * 3 = 9 [no good]
4 * 4 = 16 [no good]
5 * 5 = 2
56 * 6 = 3
67 * 7 = 49 [no good]
8 * 8 = 64 [no good]
9 * 9 = 81 [no good]
At this point we have four numbers to test. B = 0 is impossible—CB5B would have to equal 0. Similarly, B = 1 is impossible, because then the product would have to be a three digit number equal to AAB, not a four digit number.
So what if B = 5? When we multiply those 5s in the units, we get 25. So that works. Now carry the 2 next to the A. Now we know (5 * A) + 2 that again results in 5. No value of A is possible for this; well will always end up with a 7 or a 2. For example,
(5 * 2) + 2 = 12
(5 * 3) + 2 = 17
So B can't be 5. By the process of elimination, B must equal 6!
AA6 × 6 = C656
Now, given B = 6, so 6 * 6 = 36. That means (6 * A) + 3 must give us a number that ends in 5. Let's experiment,
(6 * 2) + 3 = 1
5(6 * 3) + 3 = 21 [no good]
(6 * 4) + 3 = 27 [no good]
(6 * 7) + 3 = 4
5That means that either A = 2 or A = 7. When we add that 3 (carried from 6 * 6 = 36), we get a tens digit in the product that equals 5.
It turns out that A = 2 doesn't work because of the A in the hundreds place and the B in the product. If A were 2 then B would have to be 3. For example,
(2 * 6) + 1 = 13
We already know that B must be 6 so we can eliminate 2 as a possibility. So A = 7 and we can solve the problem.
A = 7, B = 6, and C = 4, so A + B + C = 17Answer = (E)
In case you're curious, with all the numbers in place, the product is
766 × 6 = 4656
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