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02 Jan 2023, 06:44
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2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
So, 2 has a cyclicity of 4 and 86/4 = 21 with remainder 2 which means that 2^86 has 4 at the units place. When we divide 4 by 9, remainder = 4 which is our answer
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
So, 2 has a cyclicity of 4 and 86/4 = 21 with remainder 2 which means that 2^86 has 4 at the units place. When we divide 4 by 9, remainder = 4 which is our answer
02 Jan 2023, 18:02
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asmba
You've correctly found the units digit of 2^86, but we're dividing by 9 here, and a number's units digit doesn't tell you anything about the remainder you'll get when you divide by 9. Notice that 4, 14, 24, and 34 all have different remainders when you divide by 9, for example. So it's actually a coincidence that this gives you the right answer, and there's a reason earlier solutions in this thread are a bit complicated. That said the GMAT might ask for the units digit of 2^86, but it will not ask for the remainder when you divide 2^86 by 9, so the method you've used is the important one to know for the test.
19 Mar 2024, 10:07
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(2^86)/9, Remainder = ?
Sol: For any question related to remainder, we need to calculate cyclicity of that number.
Step-1: Calculate Cyclicity
(2 ^1) /9 = 2/9, Remainder = 2
(2 ^2) /9 = 4/9, Remainder = 4
(2 ^3) /9 = 8/9, Remainder = 8
(2 ^4) /9 = 16/9, Remainder = 7
(2 ^5) /9 = 32/9, Remainder = 5
(2 ^6) /9 = 64/9, Remainder = 9
(2 ^7) /9 = 128/9, Remainder = 2
(2 ^8) /9 = 256/9, Remainder = 4
So Remainder repeats after every 6th cycle, Cyclicty = 6;
Step-2: Divide the power with cyclicity
86/6 , remainder = 2
Note:
If you get remainder as 0 then take the highest cyclicity. (For example, if we get remainder 0 in this problem, we take cyclicity as 6 and answer will be 9) .
The Remainder at Cyclicity 2 is 4.
So the Remainder when (2^86)/9 is 4.
Sol: For any question related to remainder, we need to calculate cyclicity of that number.
Step-1: Calculate Cyclicity
(2 ^1) /9 = 2/9, Remainder = 2
(2 ^2) /9 = 4/9, Remainder = 4
(2 ^3) /9 = 8/9, Remainder = 8
(2 ^4) /9 = 16/9, Remainder = 7
(2 ^5) /9 = 32/9, Remainder = 5
(2 ^6) /9 = 64/9, Remainder = 9
(2 ^7) /9 = 128/9, Remainder = 2
(2 ^8) /9 = 256/9, Remainder = 4
So Remainder repeats after every 6th cycle, Cyclicty = 6;
Step-2: Divide the power with cyclicity
86/6 , remainder = 2
Note:
If you get remainder as 0 then take the highest cyclicity. (For example, if we get remainder 0 in this problem, we take cyclicity as 6 and answer will be 9) .
The Remainder at Cyclicity 2 is 4.
So the Remainder when (2^86)/9 is 4.
