Casting out the 9s

Some students may have been taught to use casting out the 9s as a way to double check their math. This number-theory based technique works as follows:

13 x 13 = 169. Is this true? Let's check by adding the digits together.
1 + 3 = 4
1 + 3 = 4
1 + 6 + 9 = 16
So to double check our answers, we could recast the problem as:

4 x 4 = 16
16 = 16 or 7=7 (1+6 = 7).

Since this is true, it is quite likely that 13x13 = 169.

Similarly does 14x14 + 15x15 = 421?

1+4 = 5
1+5 = 6
4+2+1 = 7

So we can check our answer by calculating:

Does 5x5+6x6 = 7?

25+36 = 61 and 6+1 = 7 so again, our answer is probably correct.

Why is this called casting out the 9s? Because, for example, 1+6+9 = 16 so it's faster to simply eliminate the 9. Similarly with 25+36, since 3+6 = 9 we can simply add 2+5 = 7 for a faster way to calculate a check digit.

How does this help us? Most of us do not do heavy math on the GMAT and we certainly don't waste time checking our math. What some people do not realize, however, is that this technique can be used as a solution path on math-intensive problems. Here's an example:

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A certain stock exchange designates each stock with a 1, 2, or 3 letter code, where each letter is selected from the 26 letters of the alphabet. If the letters may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes?

A. 2,951
B. 8,125
C. 15,600
D. 16,302
E. 18,278
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With an Excel spreadsheet the answer is easy. Simply calculate 26^3+26^2+26 and you have your answer. However, doing all of that math by hand will be a difficult task!

Casting out the 9s can offer us a faster way:

26 x 26 x 26 = 8 x 8 x 8 = 64 * 8 and since 6+4 = 10 our check digit is an 8.
26 x 26 = 8 x 8 = 64 = 10 = 1
26 = 8

Now we add these three check digits to get a master check digit. Since 8 + 1 = 9 our check digit becomes an 8.

Which of our answer choices sum up to 8?

A: 2951 = 2+5+1 = 8. Possible answer.
B: 8125 = (8+1) + 2+5 = 7. Eliminated.
C: 15600 = 1+5+6 = 12 and 1+2 = 3. Eliminated.
D: 16302 = 1+(6+3)+2 = 3. Eliminated.
E: 18,278 = (1+8) + (2+7) + 8 = 8. Possible answer.

Now we merely need to determine whether (A) or (E) is more likely to be the answer.

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Of course, this is not the only solution path. We could just as easily calculate that since 26x26x26 = ????6 and 26x26 = ??6, when these numbers are added together we will get:

????6 + ??6 + 26 = ????8

Since only one answer ends in 8, that must be the answer.

Alternatively, we could use an estimation path.

26 x 26 x 26 can be calculated more easily by using 20 as a reference number.

(20) 26 x 26. Since 26 is 6 more than 20, the answer will be around (26+6)20 = 32*20 = 640 and for greater accuracy we can add in the 6x6 to get the exact answer of 676, which is around 680

680 x 26 = 68x26x10
Again, we can use 20 as a reference number. Since 26 is 6 more than 20, the answer will be near (68+6)*20 = 74 * 20 = 1480 + (6*48) ≈ 6*50 so the answer must be greater than 17800 and only one answer fits the bill.

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Which solution path did you find most elegant?