What is the remainder when 7^100 is divided by 50?

Question

What is the remainder when  7^{100}  is divided by  50 ?

A. 0
B. 1
C. 7
D. 21
E. 49

Raksat · Accepted Answer

These type of questions become really simple if you understand the concept of negative remainders. Always try and reduce the dividend to 1 or -1.

= Rem

= 1
hence B

u1983 · Answer

This question essentially asking what are the last 2 digits of the expression  7^{100} ... as what ever is in the hunderds digit, if the last two are 00, the number is always divided by 50.
Now cyclicity of 7 is 4, And the numbers are 7,9,3,1........ Hence the last digit is 1 as 25*4=100
now  7^{4}  = 7*7*7*7= 2401
And the expression is  2401^{25} = in that case the last two will be always 01 ( can be tested quickly with  101^{2}  &  101^{3} )
Hence the reminder is 01.....................Hence , I would go for  option B .

FillFM · Answer

Option B

7^1  = 7
 7^2  = 49
 7^3  = 343
 7^4  = 24 01 
 7^5  = 16807
 7^6  = 117649
 7^7  = 823543
 7^8  = 57648 01

...

So, 7^100 is something that ends with 01.

The remander is 1.

MathRevolution · Answer

=>

The remainder when  7^{100}  is divided by  50  depends only on the units and tens digits.

The units digits of  7^n  cycle through the four values  7, 9, 3 , and  1 .
The tens digits of  7^n  cycle through the four values  0, 4, 4 , and  0 .

We have the following sequence of units and tens digits for  7^n :

7^1 = 07 ~ 07 
 7^2 = 49 ~ 49 
 7^3 = 343 ~ 43 
 7^4 = 2401 ~ 01 
 7^5 = 16807~ 07 
…

So,  7^{100} = (7^4)^{25}  has the same units and tens digits as  7^4 , that is,  01 .
Thus, the remainder when  7^{100}  is divided by  50  is  1 .

Therefore, B is the answer.

Answer : B

ScottTargetTestPrep · Answer

We see that 7^2 = 49, which is 50 - 1. Although 49/50 = 0 R 49, rather than using the remainder of 49, let’s call the remainder “-1”.

Since 7^100 = (7^2)^50 = 49^50, which is equivalent to (-1)^50 when it’s divided by 50, and since (-1)^50 = 1, so when (-1)^50 is divided by 50, the remainder is 1.

Answer: B

Hero8888 · Answer

7^{100}/50=49^{50}/50=(50-1)^{50}/50  - only  (-1)^{50} = 1^{50}=1  - won't be devisable by 50. The remainder is 1.

Answer B.

kanthaliya · Answer

Hi,

thanks for this solution, but I have a doubt. this question doesn't say that there is exponent for 50. then, How can we take (-1)^50 ?

Waiting for reply.

Regards,
Kishlay

anud33p · Answer

7 ^ 100 = (7 ^ 4) ^ 25 because 7 has a multiplicity of 4

7^4 = 2401 -> Express this as 2400 + 1. Therefore (2400 + 1)^25 
All terms except the last will be divisible by 50. The last term is 1. So, remainder = 1

The answer is  B

Kinshook · Answer

What is the remainder when  7^{100}  is divided by  50 ?

Remainder when 50 divides  7^{100}  
=Remainder when 50 divides  (50-1)^{50} 
= Remainder when 50 divides  (-1)^{50} 
= Remainder when 50 divides 1 = 1

IMO B

Kinshook · Answer

Asked: What is the remainder when  7^{100}  is divided by  50 ?

7^4 = 2401

7^100mod50 = 7^{4*25}mod100 = 1mod100

IMO B

GMATinsight · Answer

Remainder (7/50) = 7

Remainder ( 7^2 /50) = 49 or -1

Taking power 50 both sides

Remainder ( 7^{100} /50) =  (-1)^50 = +1

Answer: Option B

bumpbot · Answer

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What is the remainder when 7^100 is divided by 50?

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What is the remainder when 7^100 is divided by 50?

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