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27 Jun 2023, 06:00
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For a positive integer n, when 3^n is divided by 5, what is the remainder?
(1) n is a multiple of 4.
(2) n is a multiple of 2.
(1) n is a multiple of 4.
(2) n is a multiple of 2.
27 Jun 2023, 06:21
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Given: n>0
Asked: Remainder "r" when 3^n divided by 5.
Solution:
Statement 1: n = 4k.
Now we know that the cyclicity of 3 is 4. i.e. 3^1 = 3, 3^2 = 9, 3^3 = 7, 3^4 = 1, and so on.
n can be 4, 8, 12, etc. --> so 3^4k % 5 will always give the remainder as "1". eg. 3^4 % 5 = 1, 3^8 % 5 = 1 and so on. Sufficient.
Statement 2: n = 2k.
We know that the cyclicity of 2 is also 4. i.e. 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 6, and so on.
n can be 2, 4, 6, 8, etc. --> Now if n is 2 then we have a different remainder. i.e. 3^2 % 5 = 4
if n is 4, then we have a different remainder. i.e. 3^4 % 5 = 1. Hence we're not getting a single remainder value. Not Sufficient.
Answer: A
Asked: Remainder "r" when 3^n divided by 5.
Solution:
Statement 1: n = 4k.
Now we know that the cyclicity of 3 is 4. i.e. 3^1 = 3, 3^2 = 9, 3^3 = 7, 3^4 = 1, and so on.
n can be 4, 8, 12, etc. --> so 3^4k % 5 will always give the remainder as "1". eg. 3^4 % 5 = 1, 3^8 % 5 = 1 and so on. Sufficient.
Statement 2: n = 2k.
We know that the cyclicity of 2 is also 4. i.e. 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 6, and so on.
n can be 2, 4, 6, 8, etc. --> Now if n is 2 then we have a different remainder. i.e. 3^2 % 5 = 4
if n is 4, then we have a different remainder. i.e. 3^4 % 5 = 1. Hence we're not getting a single remainder value. Not Sufficient.
Answer: A
27 Jun 2023, 14:02
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Bunuel
n is a positive integer, so 3^n is a positive integer.
If an integer is divided by 5, then the remainder depends only on the units digit of the integer.
We need to answer the question:
remainder of (3^n)/5 = ?
Statement One Alone:
=> n is a multiple of 4.
n = 4q, where q is a positive integer.
3^n = 3^(4q) = (3^4)^q = 81^q
If an integer with a units digit of 1 is raised to a positive integer power, then the units digit of the resulting number is always 1. Therefore, the units digit of 3^n is 1, and the remainder in question is 1.
Statement one is sufficient. Eliminate answer choices B, C, and A.
Statement Two Alone:
=> n is a multiple of 2.
n = 2q, where q is a positive integer.
3^n = 3^(2q) = (3^2)^q = 9^q
If an integer with a units digit of 9 is raised to a positive integer power, then the units digit of the resulting number is 9 if the exponent is odd or 1 if the exponent is even. Therefore, the units digit of 3^n is 9 or 1, and the remainder in question is 4 or 1.
Statement two is not sufficient.
Answer: A
