What is the remainder if 7^10 is divided by 100?

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Sun Oct 14, 2018 4:30 am

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What is the remainder if 7^10 is divided by 100?

A] 1
B] 43
C] 19
D] 70
E] 49

OA E

Source: Manhattan Prep

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

What is the remainder if 7^10 is divided by 100?

by fskilnik@GMATH » Sun Oct 14, 2018 6:35 am

BTGmoderatorDC wrote:What is the remainder if 7^10 is divided by 100?

A] 1
B] 43
C] 19
D] 70
E] 49
Source: Manhattan Prep

First note that:

> 1 is the remainder of 101 (=1*100+1) divided by 100
> 32 is the remainder of 532 (=5*100+32) divided by 100
> 47 is the remainder of 7847 (=78*100+47) divided by 100
$${7^{10}} = K \cdot 100 + R{\mkern 1mu} {\mkern 1mu} \,\,{\mkern 1mu} {\mkern 1mu} \left( {K\,\,{\mathop{\rm int}} \,\,,\,\,\,0 \le R \le 99\,\,{\mathop{\rm int}} } \right){\mkern 1mu} $$
$$? = R$$
$${7^{10}} = {\left( {{7^2}} \right)^5} = {49^5}$$
$${49^2} = {\left( {50 - 1} \right)^2} = {5^2} \cdot {10^2} - 100 + 1 = M \cdot 100 + 1\,\,\,,\,\,\,M\,\,{\mathop{\rm int}} \ge 1\,\,\,\,\,\,\,\,\,\,\left( {M = {5^2} - 1} \right)$$
$${49^4} = {\left( {M \cdot 100 + 1} \right)^2} = {M^2} \cdot {10^4} + M \cdot 200 + 1 = N \cdot 100 + 1\,\,\,,\,\,\,\,N\,\,{\mathop{\rm int}} \,\, \ge 1\,\,\,\,\,\,\,\,\left( {N = {M^2} \cdot {{10}^2} + 2M} \right)$$
$${49^5} = \left( {N \cdot 100 + 1} \right) \cdot 49 = K \cdot 100 + 49\,\,\,,\,\,\,\,K\,\,{\mathop{\rm int}} \,\, \ge 1\,\,\,\left( {K = 49N} \right)$$
$$? = 49$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Last edited by fskilnik@GMATH on Sun Oct 14, 2018 12:03 pm, edited 1 time in total.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Ash Mo: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Wed Oct 03, 2018 10:34 am

by Ash Mo » Sun Oct 14, 2018 10:52 am

If we are asked to calculate the remainder when 6*5 is divided by 4, we observe that we can divide each of the numbers 6 and 5 and multiply their remainders to get the required answer.
For example , $$\frac{6}{4}$$ yields a remainder of 2 whereas $$\frac{5}{4}$$ gives 1.
Multiplying 2 and 1 is 2 - the same as the remainder when 30 is divided by 4.

We use this same principle in the above problem. We break the given product into small numbers whose remainder we can easily find. In this case we can write it as

$$7^9\cdot7$$ = $$343^3\cdot7$$
When 343 is divided by 100, the remainder is 43.
Our required answer is
(43*43)*(43*7)
=1849 * 301
The remainders when 1849 and 301 are divided by 100 are 49 and 1.
The product of 49 and 1 is 49 ... which is the required answer since 49 divided by 100 would continue to yield a remainder of 49.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

remainder

by GMATGuruNY » Sun Oct 14, 2018 1:56 pm

BTGmoderatorDC wrote:What is the remainder if 7^10 is divided by 100?

A] 1
B] 43
C] 19
D] 70
E] 49

When a positive integer is divided by 100, the remainder is yielded by the last two digits:
123/100 = 1 R23
548/100 = 5 R48
692/100 = 6 R92

Thus:
The remainder when 7Â¹â�° is divided by 100 is equal to the last two digits of 7Â¹â�°.

Calculate the last two digits for consecutive powers of 7 and look for a pattern:
7Â¹ --> 07
7Â² --> 49
7Â³ --> 43
7â�´ --> 01
7â�µ --> 07

The last two digits appear in a CYCLE OF 4:
07, 49, 43, 01...07, 49, 43, 01...
Implication:
When 7 is raised to a power that is a MULTIPLE OF 4 -- constituting the end of a cycle -- the last two digits will be 01.
From there, the cycle will repeat:
07, 49, 43, 01...

Since 8 is a multiple of 4, the last two digits for 7â�¸ are 01.
The cycle then repeats:
7â�¹ ---> 07
7Â¹â�° --> 49

The correct answer is E.

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Tue Oct 16, 2018 10:39 am

A general note: the GMAT will often use "remainder when divided by 10" as code for "units digit," or "remainder when divided by 100" as "the last 2 digits."

Here are more examples along those lines:
https://www.beatthegmat.com/remainder-t ... tml#767961
https://www.beatthegmat.com/if-n-and-m- ... tml#544266
https://www.beatthegmat.com/if-r-s-and- ... tml#548713
https://www.beatthegmat.com/if-a-is-an- ... 03798.html
https://www.beatthegmat.com/what-is-the ... tml#544267
https://www.beatthegmat.com/what-is-the ... tml#800962

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Tue Oct 16, 2018 5:18 pm

BTGmoderatorDC wrote:What is the remainder if 7^10 is divided by 100?

A] 1
B] 43
C] 19
D] 70
E] 49

The remainder when 7^10 is divided by 100 is equal to the last two digits of the expansion of 7^10.

Notice that 7^2 = 49, or 50 - 1. So 7^4 = (50 - 1)^2 = 2500 - 100 + 1 = 2401. The last two digits of 7^2 is 49 and those of 7^4 is 01 or simply 1. Since 7^10 = 7^2 x 7^4 x 7^4, the last two digits of 7^10 is

49 x 1 x 1 = 49

Answer: E

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Oct 16, 2018 5:26 pm

Sweeeeeeeeeeeeet solution, Scott!!

Cheers,
Brent

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Wed Oct 17, 2018 7:11 am

Brent@GMATPrepNow wrote:Sweeeeeeeeeeeeet solution, Scott!!

Cheers,
Brent

Thanks Brent!

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Founder and CEO
[email protected]

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