AAPL wrote:What is the units digit of
$$a^{4b}+2$$
where a and b are odd positive integers?
(1) a is not divisible by 5.
(2) a=x+3, where x is multiple of 10.
The OA is D.
I don't have clear this DS question. Please, can any expert assist me with it? Thanks.
(1)
a is not divisible by 5.
Since
a is not divisible by 5, a must be a number with its unit digit 1 or 3; digits 2 and 4 are excluded since a is an odd integer.
Case 1: If a = 1, then
$$a^{4b}+2 = 1^{4b}+2 = 1+2 = 3.$$ The unit digit is 3.
Case 2: If a = 3, then
a^{4b} + 2 = 3^{4b} + 2 = Unit digit 1 + 2 = 3. The unit digit is 3. Sufficient.
Note: The exponent of 3 follows a cycle of 4:
3^1 = Unit digit 3; 3^2 = Unit digit 9; 3^3 = Unit digit 7; 3^4 = Unit digit = 1.
Since the exponent of a^4b is 4b, a multiple of 4, the unit digit of a^4b must be 1.
(2) a = x + 3, where x is multiple of 10.
Since x is multiple of 10, its unit digit must be 0. Thus, the unit digit of a = 0 + 3 = 3. And the unit digit of a^{4b} + 2 = 3^{4b} + 2 = Unit digit 1 + 2 = 3. The unit digit is 3. Sufficient.
The correct answer:
D
Hope this helps!
-Jay
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