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What is the remainder, after division by 100, of 7^10 ?

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What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 07 Oct 2011, 20:45
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What is the remainder, after division by 100, of 7^10 ?

(A) 1
(B) 7
(C) 43
(D) 49
(E) 70
[Reveal] Spoiler: OA

Last edited by Bunuel on 14 Dec 2012, 01:13, edited 2 times in total.
Renamed the topic and edited the question.
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Re: PS [#permalink] New post 07 Oct 2011, 22:29
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We have such a pattern
7
7*7=49
7*7*7=343
7*7*7*7=...1
7*7*7*7*7=7
So, the last number repeats every 5th time.
10/4 = 2, and remainder is 2.
we choose 7*7
it means that 7*7*7*7*7*7*7*7*7*7=.......49
We divide by 100, it means .....,49
where 49 is a remainder.
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Re: PS: Remainder of 7^(10) divided by 100 [#permalink] New post 09 Oct 2011, 08:35
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kkalyan wrote:
What is the remainder, after division by 100, of 7^{10} ?

(A) 1
(B) 7
(C) 43
(D) 49
(E) 70



7^4 is 2401 .

So we can write 7^10 as ( 7^4 *7^4 * 7^2) divided by 100 ..

This would give us (1*1*49)/ 100 which would give remainder as 49.
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Re: PS: Remainder of 7^(10) divided by 100 [#permalink] New post 14 Dec 2012, 01:21
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thangvietnam wrote:
if we do not hav formular, how do I do?


What is the remainder, after division by 100, of 7^10 ?

(A) 1
(B) 7
(C) 43
(D) 49
(E) 70

The remainder when 7^10 is divided by 100 will be the last two digits of 7^10 (for example 123 divided by 100 yields the remainder of 23, 345 divided by 100 yields the remainder of 45).

7^{10}=(7^2)^5=49^5 --> the units digit of 49^5 will be 9 (the units digit of 9^even is 1 and the units digit of 9^odd is 9).

So, we have that 7^{10}=49^5 has the units digit of 9, thus the units digit of the remainder must also be 9. Only answer D fits.

Answer: D.
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 16 Dec 2012, 22:39
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see wrote:
Hey Bunnel,

this rule is general ? like if XXXXXX9^even the unit digit of the remainder is always 1 and XXXXX9^odd the unit digit of the remainder is always 9 ??

Thanks for your help


The units digit of 9^even is 1 and the units digit of 9^odd is 9.

If the units digit of a number is 1, then the remainder when this number will be divided by 100 will have the units digit of 1, for example 231 divided by 100 gives the reminder of 31.

If the units digit of a number is 9, then the remainder when this number will be divided by 100 will have the units digit of 9, for example 239 divided by 100 gives the reminder of 39.
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Re: PS [#permalink] New post 09 Oct 2011, 05:02
kkalyan wrote:
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Help End Exam Review Section

What is the remainder, after division by 100, of 7^{10} ?

1
7
43
49
70

PLZ EXPLAIN THE TRICK IN SOLVING THIS TYPE REMAINDER PROBLEM


Using Binomial theorem, last two digits of an exponent can be found as
7^(10)=7^(2*5)=49^5=(-1+50)^5=(-1)^5+5*(-1)^4*50=-1+50(Just considered last 2-digit of the product)=49

Look for Karishma's blogs. You may find more.

Ans: "D"
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Re: PS: Remainder of 7^(10) divided by 100 [#permalink] New post 14 Dec 2012, 00:48
if we do not hav formular, how do I do?
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 16 Dec 2012, 12:56
Hey Bunnel,

this rule is general ? like if XXXXXX9^even the unit digit of the remainder is always 1 and XXXXX9^odd the unit digit of the remainder is always 9 ??

Thanks for your help
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 17 Dec 2012, 23:14
what if the answer choise has another value with units digit 9? how do we need to proceed in that case?
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 18 Dec 2012, 00:38
Ans:

7^10 can be written as 49^5 which can be written as (49^2)^2. 49 when divided by 100 it will give a remainder of (1)^2.49=49 answer (D).
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 04 Feb 2013, 03:11
7 ^1 has last digit is 7
7^2 has last digit is 9
3
1

the last digit of 7^10 must be 9

the remainder must has the same last digit

only D fits
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 17 Dec 2013, 05:59
priyamne wrote:
Ans:

7^10 can be written as 49^5 which can be written as (49^2)^2. 49 when divided by 100 it will give a remainder of (1)^2.49=49 answer (D).


49^5 is not (49^2)^2

Would you please explain yourself. I'm trying to follow your approach but don't quite get it

Thanks
Cheers!
J :)
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Re: What is the remainder, after division by 100, of 7^10 ? [#permalink] New post 01 May 2014, 02:10
Amateur wrote:
what if the answer choise has another value with units digit 9? how do we need to proceed in that case?



Good question. Any clues Bunnuel
Re: What is the remainder, after division by 100, of 7^10 ?   [#permalink] 01 May 2014, 02:10
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