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
What is the remainder if 7^10 is divided by 100?
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First note that: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
> 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.
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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.
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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When a positive integer is divided by 100, the remainder is yielded by the last two digits:BTGmoderatorDC wrote:What is the remainder if 7^10 is divided by 100?
A] 1
B] 43
C] 19
D] 70
E] 49
123/100 = 1 R23
548/100 = 5 R48
692/100 = 6 R92
Thus:
The remainder when 7¹� is divided by 100 is equal to the last two digits of 7¹�.
Calculate the last two digits for consecutive powers of 7 and look for a pattern:
7¹ --> 07
7² --> 49
7³ --> 43
7� --> 01
7� --> 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 7� are 01.
The cycle then repeats:
7� ---> 07
7¹� --> 49
The correct answer is E.
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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
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
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The remainder when 7^10 is divided by 100 is equal to the last two digits of the expansion of 7^10.BTGmoderatorDC wrote:What is the remainder if 7^10 is divided by 100?
A] 1
B] 43
C] 19
D] 70
E] 49
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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Thanks Brent!Brent@GMATPrepNow wrote:Sweeeeeeeeeeeeet solution, Scott!!
Cheers,
Brent
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