A division problem - brain teaser?

Question

A, B, and C are different digits, each of AB and 7C is a 2-digit number, and each blank space represents a missing digit. What is the value of each of A, B, and C?

____7 C__
AB) _ _ _ _
_ _
______
_ _ _
_ 2 _
_____
0

Note that you will need to write the problem to reflect the proper spacing, e.g. the 7 in the quotient should be above the 3rd digit of teh dividend.

Please explain how you arrived at your answer.

Answer 1

I added some letters to the other spaces to help explain the solution.
.............. 7 C
..... ------------
A B ) d e f g
........... h i
.......... -------
........... j k l
........... m 2 n
.......... -------
................. 0

First, you are subtracting a two digit number (hi) from a three digit number (def). That means that you have to borrow. Thus d must be 1. Also, the last subtraction results in zero, so k must be 2 and jkl = m2n.

.............. 7 C
..... ------------
A B ) 1 e f g
........... h i
.......... -------
........... j 2 l
........... j 2 l
.......... -------
................. 0

Next you are multiplying 7 times a two digit number (AB) and getting a two digit number... the choices are:
7 x 10 = 70, 7 x 11 = 77, 7 x 12 = 84, 7 x 13 = 91, 7 x 14 = 98.

Thus A must be 1.

.............. 7 C
..... ------------
1 B ) 1 e f g
........... h i
.......... -------
........... j 2 l
........... j 2 l
.......... -------
................. 0

For the second multiplication, C must be bigger than 7, because you get a *three* digit result. Our choices for AB are 10, 11, 12, 13 or 14.

When you try 8 times these numbers you get:
8 x 10 = 80
8 x 11 = 88
8 x 12 = 96
8 x 13 = 104
8 x 14 = 112
None of these have a 2 in the middle, so C must be 9.

.............. 7 9
..... ------------
1 B ) 1 e f g
........... h i
.......... -------
........... j 2 l
........... j 2 l
.......... -------
................. 0

Trying 9 times these numbers you get:
9 x 10 = 90
9 x 11 = 99
9 x 12 = 108
9 x 13 = 117
9 x 14 = 126
The only solution that works is 9 x 14 since it has a 2 in the middle. So B = 9.

.............. 7 9
..... ------------
1 4 ) 1 e f g
........... h i
.......... -------
........... 1 2 6
........... 1 2 6
.......... -------
................. 0

Now you can figure out the rest easily. Here's the complete equation:

.............. 7 9
..... ------------
1 4 ) 1 1 0 6
........... 9 8
.......... -------
........... 1 2 6
........... 1 2 6
.......... -------
................. 0

So 1106 / 14 = 79

A = 1, B = 4, C = 9

Answer 2

First, we see that A=1 because the product of 7 x AB is a 2-digit number; this wouldn't be possible for A>1.

Seeing that 7 x AB gives a 2-digit result and C x AB gives a 3-digit result, we know C > 7, i.e. 8 or 9.

We know B is not 0; if B=0, you can't get a 3-digit number multiplying C x AB. In fact, we can narrow the possible values for B to: { 2, 3, 4 } If B were 5 or more, 7 x AB would not be a 2-digit number.

So we have: AB is one of {12, 13, 14} and C is one of { 8, 9 }

Last step: which combination of values for A, B and C gives you a 3-digit number with a middle digit of '2' when you multiply C x AB?

Just try them all:
12 x 8 = 96
13 x 8 = 104
14 x 8 = 112
12 x 9 = 108
13 x 9 = 117
14 x 9 = 126

The last one is the only one that fits, and we have our answer: A=1, B=4, C=9.