Manhattan GMAT Challenge Problem of the Week - 28 Sept 2011

by Manhattan Prep, Sep 28, 2011

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Question

Starting with 0, a mathematician labels every non-negative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power?

A. alpha

B. beta

C. gamma

D. delta

E. epsilon

Answer

Since there are five labels, given in order to all the integers, the label alpha is given to 0, 5, 10, etc. that is, the alphas are the multiples of 5 and end in 0 or 5. All the other labels correspond to non-multiples of 5 in fact, they each correspond to particular remainders and particular units digits. For instance, the betas (1, 6, 11, 16, etc.), which all end in 1 or 6, also all leave a remainder of 1 after division by 5. The gammas correspond to a remainder of 2 (units digits = 2 or 7). Deltas correspond to a remainder of 3 (units digits = 3 or 8), and epsilons correspond to a remainder of 4 (units digits = 4 or 9).

Now, a gamma raised to the seventh power will be large, even if we pick the smallest gamma (2 itself). But all we need is the units digit of the result. So compute the units digit in stages:

First power: units digit = 2

Second power: units digit = 22 = 4

Third power: units digit = 24 = 8 (remainder = 3)

Fourth power: units digit = 28 = 16 = 6 (units digit only) (remainder = 1)

Fifth power: units digit = 26 = 12 = 2 (units digit only) (remainder = 2)

Sixth power: units digit = 22 = 4 (remainder = 4)

Seventh power: units digit = 24 = 8 (remainder = 3)

Do the same for the delta.

First power: units digit = 3

Second power: units digit = 33 = 9 (remainder = 4)

Third power: units digit = 39 = 27 = 7 (remainder = 2)

Fourth power: units digit = 37 = 21 = 1 (remainder = 1)

Fifth power: units digit = 31 = 3 (remainder = 3)

Sixth power: units digit = 33 = 9 (remainder = 4)

Seventh power: units digit = 39 = 27 = 7 (remainder = 2)

[pmath]G a m m a^7[/pmath] gives us a remainder of 3. [pmath]D e l t a^7[/pmath] gives us a remainder of 2. Adding the remainders, we get a remainder of 5, which is the same as a remainder of 0 (remember, were talking about division by 5).

So the sum gets a label of alpha.

The correct answer is A.

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