If n is a positive integer, what is the remainder when (8^4n+2)(6^n)

Question

If n is a positive integer, which of the following is the remainder of  \frac{(8^{4n+2})(6^n)}{10}  ?

A. 4
B. 5
C. 6
D. 7
E. 8

narenbs · Answer

/10 
 /10
 /10
 /10
2^12n has a units digit of 6, because the powers of 2 follow cyclicity of 4 and 12 is a multiple of 4. 2^6 has a units digit of 4. And 6^n always has a units digit of 6 (regardless of the integer value of n).

From the units digits of 6, 4 and 6, we know that the units digit of the numerator is 4. Therefore, when divided by 10, the remainder is 4.

sivatx2 · Answer

Unit digit of power of 8 repeats every 4 times - 8,4,2,6,8,4,2,6,

So the unit digit on the 8 to the power of (4n+2) will have the unit digit as 4.
Irrespective of the value of n, 6 to the power any number will have a unit digit 6.

So multiplying 8 to the power of (4n+2) and 6 to the power of n will have the unit digit (4x6) = 4.

So dividing the product of the above powers, with 10, will result in with a reminder of 4.

So A is the correct answer.

Yogananda · Answer

8 to the power 4n+2 will always provide unit digit as 4, try putting different values for n.
So,when divided by 10 the remainder would be 4.

Posted from my mobile device

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=B6QU1StB544

Remainder of  (8^{4n+2})(6^n)  by 10 = Unit's digit of  (8^{4n+2})(6^n)

To find the  Unit's digit of   (8^{4n+2})    we will find the cycle of unit's digit of power of 8

Unit's digit of  8^1  = 8
Unit's digit of  8^2  = 4
Unit's digit of  8^3  = 2
Unit's digit of  8^4  = 6
Unit's digit of  8^5  = 8

=> We have a cycle of 4
=> We will divide 4n + 2 by 4 and check the remainder
Since Remainder is 2
=> Unit's digit of   (8^{4n+2})   = Unit's digit of  8^2  = 4

=>  Remainder of  (8^{4n+2})(6^n)  by 10 = 4

So,  Answer will be A 
Hope it helps!

Remainders by 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

ManifestDreamMBA · Answer

We just need to find the units digit to answer this
Given n is a positive integer, 6^n will ends in 6

8^{4n+2} 
 2^{6(2n+1)} 
 4^{3(2n+1)} 
3(2n+1) will be odd and 4^odd ends in 4

So, the units digits of  (8^{4n+2})(6^n)  = 4*6 = 4

If n is a positive integer, what is the remainder when (8^4n+2)(6^n)

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GMAT Problem Solving (PS) Questions

If n is a positive integer, what is the remainder when (8^4n+2)(6^n)

Prep Toolkit

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My Rewards