Problem Solving

M7MBA wrote: ↑

Thu Sep 17, 2020 1:13 am

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: A

Solution:

Notice that an alpha number is a number with units digit of 0 or 5, beta 1 or 6, gamma 2 or 7, delta 3 or 8, and epsilon 4 or 9. That is, we can determine the designated “name” of a number by its units digit.

Let’s let the gamma number be 2 (the smallest gamma number) and the delta number be 3 (the smallest delta number). Recalling the units digit pattern of powers of 2 is 2-4-8-6, we see that the units digit of 2^7 is 8. Similarly, recalling that the units digit pattern of powers of 3 is 3-9-7-1, we see that the units digit of 2^7 is 7. Since 8 + 7 = 15, the units digit of 2^7 + 3^7 is 5, which is an alpha number.

Answer: A