Bunuel wrote:
If x is a positive integer, what is the units digit of \(24^{5 + 2x}*36^6*17^3\)?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 8
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:a.
Unknowns: Units digits of 24^(5 + 2x), 36^6, and 17^3.
Given: We only need the units digit of the product, not the value. x is a positive integer.
Constraints: If x is a positive integer, 2x is even, and 5 + 2x must be odd. Units digit of a product depends only on the units digit of multiplied numbers.
Question: What is the units digit of the product 24^(5 + 2x)*36^6*17^3?
b. Find/Recall the pattern for units digits → use the Unit Digit Shortcut
c. Units digit of 24^(5 + 2x) = units digit of 4^odd. The pattern for the units digit of 4^(integer) = [4, 6]. Thus, the units digit is 4.
Units digit of 36^6 must be 6, as every power of 6 ends in 6.
Units digit of 1^3 = units digit of 7^3. The pattern for the units digit of 7 integer = [7, 9, 3, 1]. Thus, the units digit is 3.
The product of the units digits is (4)(6)(3) = 72, which has a units digit of 2. The answer is A.
d. Patterns were very important on this one! If we had forgotten any of the patterns, we could just list at least the first four powers of 4, 6, and 7 to recreate them.
I don't believe in giving up!