What is the remainder when 13^17 + 17^13 is divided by 10?

Question

What is the remainder when  13^{17} + 17^{13}  is divided by 10?

A. 0
B. 2
C. 4
D. 6
E. 8

Chethan92 · Answer

13^17/10 = 3^17/10 = (3^2)^8*3/10 = 9^8*3/10 = (-1)^8*3/10 = 3.

17^13/10 = 7^13/10 = (7^2)^6*7/10 = (49)^6*7/10 = (-1)^6*7/10 = 7.

Remainder = (3+7)/10 = 0.

A is the answer

Posted from my mobile device

hibobotamuss · Answer

Hi  Afc0892  could you explain how you broke it down?

Chethan92 · Answer

Hey  hibobotamuss , sure.

First, take  13^{17} . When 13 is divided by 10, the remainder is 3.
Now  3^{17} . Now  3^2  = 9. When 9 is divided by 10, the remainder is either 9 or -1.
 3^{17}  =  \frac{(3^2)8*3}{10} . -1^8 is 1. then the remainder left is 3.

Similarly doing this for 17^13, we'll get 7 as the remainder.
Sum of the remainders is 10. hence when 10 is divided by 10, the remainder is 0.

Hope it's clear.

PS: I think there is a direct formula to calculate, but I don't remember it.

stesmit · Answer

13^17 -->  3^17 
17^13 -->  7^13

The returing digits:
3 =   So a multiple of 4
7 =   Also a multiple of 4

3^16 = 1 hence 3^17 should be 3 --> The next number in the sequence
7^12 = 1 hence 7^13 should be 7 --> The next number in the sequence

(3+7)/10= 0

A

ArjunJag1328 · Answer

\frac{13^17 + 17^13}{10}

13^1 = 13 
13^2 = 169 
13^3 = 2197 
13^4 = 28561

17^1 = 17
17^2 = 289
17^3 = 4913
17^4 = 83521

From the above chart, we can find out the unit digits of 13^17 and 17^13 and in fact this information would be sufficient for us to answer the question.
13^17's unit digit will be 3
17^13's unit digit will be 7

The unit digit will be '0' when you add 13^17 + 17^13 and therefore any number that ends with a '0' is always divisible by 10.

Hence the answer IMO is (A) 0

EncounterGMAT · Answer

The pattern for the units digits of 3 is   => 3^17 must end in 3.
The pattern for the units digits of 7 is   => 7^13 must end in 7.

The sum of both these units digits is 3 + 7 = 10. Thus, the units digit is 0. Any number that ends with 0 is completely divisible by 10. Hence remainder is zero.

IMO answer is option A.

Archit3110 · Answer

took <1 min ; solve using cyclicity rule 
3^1:3
3^2:9
3^3:7
3^4:1

similarly for 7^1:7
7^2: 9
7^3:3
7^4:1

both have '4' as cyclicity

so for  13^{17} + 17^{13}

can be written as ( 10+3)^17 + (10+7)^13

anything to power of 10 would always give units digit as '0'

so for 3^17+7^13
 ' units digit ' : 3^17: 3 and 7^13: 7

or we can say that unit digit of expression ( 10+3)^17 + (10+7)^13 would be 0+7+3= 0 so remainder would be 0 when divided by 10 ... 
IMO A 
hope this helps..

ScottTargetTestPrep · Answer

Solution:
 
The remainder when a number is divided by 10 is equal to the units digit of the number. Therefore, we need to determine the units digit of 13^17 + 17^13. We need to determine the units digit of 13^17 and the units digit of 17^13 and then add the two units digits.

Recall that the units digit pattern of powers of 3 (or numbers with units digit 3) is 3-9-7-1. Therefore, the units digit of 13^16 is 1, and that of 13^17 is 3. Similarly, the units digit pattern of powers of 7 (or numbers with units digit 7) is 7-9-3-1. Therefore, the units digit of 17^12 is 1, and that of 17^13 is 7.

Since 3 + 7 = 10, the units digit of 13^17 + 17^13 is 0.

Answer: A

BrushMyQuant · Answer

We need to find what is the remainder when  13^{17} + 17^{13}  is divided by 10

Theory: Remainder of sum of two numbers = Sum of their individual remainders
Remainder of any number by 10 = Unit's digit of that number

=> Remainder of  13^{17} + 17^{13}  by 10 = Remainder of  13^{17}  by 10 + Remainder of  17^{13}  by 10

Unit's digit of  13^{17}

= Unit's digit of  3^{17}

We can do this by finding the pattern / cycle of unit's digit of power of 3 and then generalizing it.

Unit's digit of  3^1  = 3
Unit's digit of  3^2  = 9
Unit's digit of  3^3  = 7
Unit's digit of  3^4  = 1
Unit's digit of  3^5  = 3

So, unit's digit of power of 3 repeats after every  4^{th}  number.
=> We need to divided 17 by 4 and check what is the remainder
=> 17 divided by 4 gives 1 remainder

=>  3^{17}  will have the same unit's digit as  3^1  = 3
=> Unit's digits of  13^{17}  = 3

Unit's digit of  17^{13}

= Unit's digit of  7^{13}

We can do this by finding the pattern / cycle of unit's digit of power of 7 and then generalizing it.

Unit's digit of  7^1  = 7
Unit's digit of  7^2  = 9
Unit's digit of  7^3  = 3
Unit's digit of  7^4  = 1
Unit's digit of  7^5  = 7

So, unit's digit of power of 7 repeats after every  4^{th}  number.
=> We need to divided 13 by 4 and check what is the remainder
=> 13 divided by 4 gives 1 remainder

=>  7^{13}  will have the same unit's digit as  7^1  = 7
=> Unit's digits of  17^{13}  = 7

=> Unit's digits of  13^{17}  + Unit's digits of  17^{13}  = 3 + 7 = 10

But remainder of  13^{17} + 17^{13}  by 10 cannot be more than or equal to 10
=> Remainder = Remainder of 10 by 10 = 0

So,  Answer will be A 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the remainder when 13^17 + 17^13 is divided by 10?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the remainder when 13^17 + 17^13 is divided by 10?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards