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Re: What is the remainder when 2^20 is divided by 10 ?
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06 May 2016, 01:50
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broilerc wrote:
What is the remainder when 2^20 is divided by 10 ?
A. 0 B. 2 C. 4 D. 6 E. 8
Hi,
TWO ways-
1) Cyclic pattern of units digit- Remainder when div by 10 is nothing BUT units digit \(2^1 = 2......... 2^2 = 4.......... 2^3 = 8.......... 2^4 = 16.. or... 6........\) and this carries on in same pattern.... 2, 4, 8, 6, 2, 4, 8, 6... so 20 is div by 4.. so 2^20 will have UNITS digit same as 4th power.. ans 6
2) binomial expansion \(2^{20} = (2^5)^4 = 32^4 = (30+2)^4\).. Now the above expression will have all other terms div by 10 except 2^4... \(2^4 = 16\).. and 16 div by 10 gives a remainder of 6..
Re: What is the remainder when 2^20 is divided by 10 ?
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06 May 2016, 04:32
chetan2u wrote:
broilerc wrote:
What is the remainder when 2^20 is divided by 10 ?
A. 0 B. 2 C. 4 D. 6 E. 8
Hi,
TWO ways-
1) Cyclic pattern of units digit- Remainder when div by 10 is nothing BUT units digit \(2^1 = 2......... 2^2 = 4.......... 2^3 = 8.......... 2^4 = 16.. or... 6........\) and this carries on in same pattern.... 2, 4, 8, 6, 2, 4, 8, 6... so 20 is div by 4.. so 2^20 will have UNITS digit same as 4th power.. ans 6
2) binomial expansion \(2^{20} = (2^5)^4 = 32^4 = (30+2)^4\).. Now the above expression will have all other terms div by 10 except 2^4... \(2^4 = 16\).. and 16 div by 10 gives a remainder of 6..
Re: What is the remainder when 2^20 is divided by 10 ?
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06 May 2016, 10:33
[quote="broilerc"]What is the remainder when 2^20 is divided by 10 ?
A. 0 B. 2 C. 4 D. 6 E. 8
2^20 can be written as (2^4)^5
16^5=16*16*16*16*16
any no having unit digit 6 multiplied n no. of times will allways give you unit digit 6. so the unit digit of this no will be 6. any no. That is divided by 10 will allways give you unit digit as a reminder.
PS: Its better to avoid this approach during actual GMAT exam, just posting an alternate approach for educational purpose.
How come you can't just take 2^19/5 and say that it is remainder 3? Isn't 2^20/10 = 2^19/5? Yet you get a remainder of 6 for the first one and a 3 for the second. I'm confused how you knew to multiply the remainder of 2^18/5 and 2/5.
PS: Its better to avoid this approach during actual GMAT exam, just posting an alternate approach for educational purpose.
How come you can't just take 2^19/5 and say that it is remainder 3? Isn't 2^20/10 = 2^19/5? Yet you get a remainder of 6 for the first one and a 3 for the second. I'm confused how you knew to multiply the remainder of 2^18/5 and 2/5.
Simplification changes the remainder. Look at this:
25/10 - Remainder 5 But 5/2 - Remainder 1
Dividend = Quotient * Divisor + Remainder
Like in the example above, when dividend and divisor are divided by 5, the Remainder gets divided by 5 too. So to get the actual Remainder, you need to multiply the Remainder by 5 again.
Hence when you use 2^19/5 (after dividing both Dividend and Divisor by 2) and get the remainder 3, you need to multiply it by 2 back to get the remainder 6.
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What is the remainder when 2^20 is divided by 10 ?
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01 Jan 2017, 14:15
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Easy explanation:
\(2^{20}=4^{10}=16^5\)
All powers of 6 end with a 6 in the units digit, so \(16^5\) must also end with a 6. Thus, when divided by 10, the remainder must be 6.
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Affiliations: AB, cum laude, Harvard University (Class of '02)
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What is the remainder when 2^20 is divided by 10 ?
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12 Jan 2017, 18:21
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ziyuenlau wrote:
mcelroytutoring wrote:
Easy explanation:
\(2^{20}=4^{10}=16^5\)
All powers of 6 end with a 6 in the units digit, so \(16^5\) must also end with a 6. Thus, when divided by 10, the remainder must be 6.
mcelroytutoring Could you briefly explain this? I am not understand about this.
Sure! When you multiply 20 2s together, you get 10 4s because each pair of 2s makes a 4. (2 x 2 = 4) When you multiply 10 4s together, you get 5 16s because every pair of 4s makes a 16 (4 x 4 = 16).
Every power of 6 ends with a 6 because \((6)(6) =36\) and \((6)(6)(6) = (36)(6) = 216\) and \((6)(6)(6)(6) = 1296\), etc. And the remainder when you divide by 10 will always be equal to 6, because multiples of 10 always end in 0. \(36/10 = 3 R 6, 216/10 = 21 R 6, 1296/10 = 129 R 6\), etc.
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Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching worldwide since 2002.
One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong).
You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y7knw7bt Date of Birth: 09 December 1979.
Here the power of 2 is 20. As the cyclicity of 2 is 4, we will divide 20 by 4 which gives us a remainder of 0. Thus the unit's digit (here) would be 6.
Therefore the remainder upon division with 10 is 6.
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Re: What is the remainder when 2^20 is divided by 10 ?
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06 Apr 2017, 09:41
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broilerc wrote:
What is the remainder when 2^20 is divided by 10 ?
A. 0 B. 2 C. 4 D. 6 E. 8
We need to determine the remainder when 2^20 is divided by 10. To do so, recall that any number divided by 10 will produce the same remainder as the units digit of that number. Thus, let’s determine the units digit of 2^20. The pattern of units digits of 2 when raised to a positive integer exponent is:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6
2^5 = 2
We see that the pattern is 2-4-8-6. Furthermore, 2^4k, in which k is a positive integer, will always have a units digit of 6.
Thus, the units digit of 2^20 is 6. Dividing 6 by 10 yields a remainder of 6; thus, dividing 2^20 by 10 also yields a remainder of 6.
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67% (00:58) correct 
