In the multiplication problem below, the letters G, M, A, T, and H represent different digits. What is the value of G+M+A+T+H?
I don't consider myself a 'GMAT expert', but couldn't pass this one up. Good question! Just don't tell us you saw this on a GMAT test.
Here we go:
2008 x HT = 2000 X T + 8 X T + 2000 X H0 + 8 X H0
This can actually be written as:
2000 X T + 8 X T
(+) 2000 X H0 + 8 X H0
Know what the sum of the digits is of the sum of individual numbers? Its nothing but the sum of the digits of the individual numbers (with no repeats)!
So G+M+A+T+H = 2+T+8+H = 10+T+H
I'll use an example to illustrate if my explanation was too cryptic.
The trouble with alphabet multiplication is that the normal way we all know to multiply doesn't work, because we don't know the carryovers to transfer to the next digit (on multiplication first and then the addition). But you don't need to know the carryovers if you go back to basics of what multiplication of two numbers really is. Consider this example:
45 X 63. Now assume you couldn't solve this the normal way, i.e. no carryovers allowed in multiplication or addition.
You would break 45 X 63 = (40 + 5) X ( 60 + 3) = 40 X 60 + 40 X 3 + 60 X 5 + 5 X 3
This is nothing but:
40 X 3 + 5 X 3
(+) 40 X 60 + 60 X 5
Look familiar? This is the breakup you would get if you broke up each of the two lines if you were to multiply and if you could use carryovers!
Now we are left with addition. The thing to note here is that whenever you add any set of numbers, the sum of the digits of the resultant number is equal to the sum of the individual digits of each of the numbers.
So if you were to add:
135 + 2700 = 2835, notice 1+3+5+2+7 = 18 = 2+8+3+5. This isn't magic. When you carry over in addition, you are transferring an amount from one number to the next digit to add to, but because the gain-loss is the same, in the end you land up with the same number when you add the individual digits.
This question totally baked my noodles. Thanks for sharing.
Hope this helped.
The only ability the GMAT is an indicator of...is the ability to do well on the GMAT.