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What is the remainder when 5^16 - 3^16 is divided by 8?

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e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2460
What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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19 Dec 2018, 10:18
00:00

Difficulty:

15% (low)

Question Stats:

82% (01:19) correct 18% (01:23) wrong based on 85 sessions

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What is the remainder when $$5^{16} - 3^{16}$$ is divided by 16?

A. 0
B. 1
C. 3
D. 5
E. 7

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Re: What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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19 Dec 2018, 10:50
1
EgmatQuantExpert wrote:
What is the remainder when $$5^{16} - 3^{16}$$ is divided by 16?

A. 0
B. 1
C. 3
D. 5
E. 7

To read all our articles:Must read articles to reach Q51

$$5^{16} - 3^{16}$$

$$5^8*2 - 3^8*2$$

$$(5^8 +3^8) (5^8 - 3^8)$$

now just break down 5^8 - 3^8.

apply $$a^2 -b^2.$$

at least we got : ($$5^8 +3^8) (5^4 +3^4) (5^2 +3^2) ( 5 +3 ) (5-3)$$

Now divide :

$$(5^8 +3^8) (5^4 +3^4) (5^2 +3^2) ( 5 +3 ) (5-3) /16.$$

$$(5^8 +3^8) (5^4 +3^4) (5^2 +3^2) *8*2 /16.$$

Reminder is 0.

A is the correct answer.
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Re: What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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20 Dec 2018, 00:25
I solved using cyclicity

for any power to 5 we would always get 5 as units digit and for 3 ^ 16 we would get 1 as unit digits

so 5-1= 4 unit digit

16= 2^4

for an even Nr ending with 4 unit digit when divided by 2 would always give remainder as 0 , so IMO A ..

EgmatQuantExpert wrote:
What is the remainder when $$5^{16} - 3^{16}$$ is divided by 16?

A. 0
B. 1
C. 3
D. 5
E. 7

To read all our articles:Must read articles to reach Q51

_________________

If you liked my solution then please give Kudos. Kudos encourage active discussions.

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2460
What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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01 Jan 2019, 21:56

Solution

To find:
We are asked to find out,
• The remainder when $$5^{16} – 3^{16}$$ is divided by 8

Approach and Working:
• $$5^{16} – 3^{16}$$ can be written as $$(5^8)^2 – (3^8)^2$$

We know that, $$a^2 – b^2 = (a + b) * (a - b)$$
• Thus, $$(5^8)^2 – (3^8)^2 = (5^8 + 3^8) * (5^8 – 3^8) = (5^8 + 3^8) * (5^4 + 3^4) * (5^4 – 3^4) = (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5^2 - 3^2) = (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5 + 3) * (5 – 3)$$
• So, $$5^{16} – 3^{16} = 16 * (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5^2 - 3^2)$$

If you observe carefully, the above expression is a multiple of 8
• Therefore, the remainder will be 0

Hence the correct answer is Option A.

_________________

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Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

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What is the remainder when 5^16 - 3^16 is divided by 8? &nbs [#permalink] 01 Jan 2019, 21:56
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