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Remainder Problem

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Remainder Problem

by MakeitHappen » Wed Apr 21, 2010 10:15 am
I found the below mentioned tip for solving remainder problem from some forum(cant rem which one but i had it saved on my laptop..so i dont rem the source):
Cycle of powers : This is used to find the remainder of , when divided by 10, as it helps us in figuring out the last digit of .

The cycle of powers for numbers from 2 to 10 is given below:

2: 2, 4, 8, 6 → all will have the same last digit.

3: 3, 9, 7, 1 → all will have the same last digit.

4: 4, 6 → all will have the same last digit.

5: 5 → all will have the same last digit.

6: 6 → all will have the same last digit.

7: 7, 9, 3, 1 → all will have the same last digit.

8: 8, 4, 2, 6 → all will have the same last digit.

9: 9, 1 → all will have the same last digit.

10: 0 → all will have the same last digit.

I didnt quite understand the above!!! Any help appreciated...thanks in advance
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by ajith » Wed Apr 21, 2010 10:41 pm
Moved to GMAT Math Forum!
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by Brian@VeritasPrep » Thu Apr 22, 2010 8:54 am
Hey, MakeitHappen (love the positive vibes from the username!),

The concept behind what you posted is hugely helpful, but it's not necessary (or recommended) to memorize that list. I'll break the importance into two parts:

1) Exponents are very pattern driven! That list you have below shows the cycles for the units digits of exponents. Take 9, for example. If you were to take 9 to a series of exponents, you'd get:

9^1 = 9
9^2 = 81
9^3 = 729

but from here trying to keep doing the multiplication gets messy with so many digits. However, just multiplying the units digits makes it easy to get a glimpse of what the result will look like:

9^4 will equal 729 * 9, and for the units spot 9*9 will yield a 1, so if we just look at the units digits we'll find that:

9^1 yields a units digit of 9
9^2 --> 1
9^3 --> 9
9^4 --> 1

And we see that there's a pattern. There are only two unique units digits that can arise when taking 9 to an exponent (an integer > 0), and the cycle demonstrates that every even exponent will yield a 1 and every odd exponent will yield a 9.

The rest of that table you posted shows the cycles for every other exponent, and they can run in cycles of 1 (e.g. 6, for which the units digit will always be a 6); 2 (like we saw with 9); and 4 (see the table for bases 7, 4, and others...and, more importantly, try to prove that pattern to yourself so that you can see why it holds).

Because, when you're looking at these exponential cycles, the most math you ever have to do is 4 quick multiplication problems, you don't need to memorize that table - just know that for whichever base you have, you can establish that pattern by testing the smallest exponents until you find a repeat.


2) This pattern demonstrates an incredibly important GMAT strategy when you face math problems that appear to require extensive calculation: Find the Pattern!

If you can establish a pattern using small numbers and prove that it will hold as the values increase, then you can solve big problems much more efficiently than you could by actually grinding through the math. When a problem appears to require extensive calculation, see if you can find a pattern with the types of numbers given to you, and that may be your key to unlocking the problem. As mentioned above, exponents, in particular, tend to be quite pattern-driven, so that's a problem type that lends itself quite nicely to this strategy.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep

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