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12 Feb 2017, 02:32
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If n is an integer, what is the unit digit of 2^n?
a) n is a multiple of 6
b) n is a multiple of 4
In here answer is [b]. But I have little confusion about this. would you like to explain me. please
Source: Math Revolution
a) n is a multiple of 6
b) n is a multiple of 4
In here answer is [b]. But I have little confusion about this. would you like to explain me. please
Source: Math Revolution
Vaidya
Joined: 02 Oct 2016
Last visit: 10 Sep 2025
Posts: 32
Given Kudos: 20
Location: India
Schools: HEC Dec '17 (D) IE (A) INSEAD Jan'23 intake
GMAT 1: 740 Q49 V42
12 Feb 2017, 02:48
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Hi Farjin,
Unit digits of different powers of the integer 2 follow a specific cycle.
2^1 will have 2 as the units digit
2^2 will have 4
2^3 will have 8
2^4 will have 6
2^5 will have 2
2^6 will have 4
2^7 will have 8
2^8 will have 6.
If you look carefully, the cycle 2,4,8,6 is repetitive.
When you can break down the power to the form 4m, 4m+1, 4m+2, 4m+3, where m is any integer, you'll be able to find the units digit.
For example, 2^21 = 2^(20+1) = 2^ [4(5)+1]. In this case, you can ignore the part which is a multiple of 4 and focus on the +1 part.
Hence 2^1 will have the same units digit as 2^21.
Coming to your question,
Statement 1 says n is a multiple of 6.
Case 1: n = 6
2^6 will have 4 as the unit digit.
Case 2: n = 12
2^12 will have 6 as the unit digit.
You now have two values, so answer isn't unique.
Statement 2 says n is a multiple of 4
N can be 4, 8, 12, 16, 20....
Whatever the value of n, 2^n will have 6 as the units digit. Hence, the answer is unique and sufficient.
Unit digits of different powers of the integer 2 follow a specific cycle.
2^1 will have 2 as the units digit
2^2 will have 4
2^3 will have 8
2^4 will have 6
2^5 will have 2
2^6 will have 4
2^7 will have 8
2^8 will have 6.
If you look carefully, the cycle 2,4,8,6 is repetitive.
When you can break down the power to the form 4m, 4m+1, 4m+2, 4m+3, where m is any integer, you'll be able to find the units digit.
For example, 2^21 = 2^(20+1) = 2^ [4(5)+1]. In this case, you can ignore the part which is a multiple of 4 and focus on the +1 part.
Hence 2^1 will have the same units digit as 2^21.
Coming to your question,
Statement 1 says n is a multiple of 6.
Case 1: n = 6
2^6 will have 4 as the unit digit.
Case 2: n = 12
2^12 will have 6 as the unit digit.
You now have two values, so answer isn't unique.
Statement 2 says n is a multiple of 4
N can be 4, 8, 12, 16, 20....
Whatever the value of n, 2^n will have 6 as the units digit. Hence, the answer is unique and sufficient.
