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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: 3019
What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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19 Dec 2018, 11:18
1
10
00:00

Difficulty:

25% (medium)

Question Stats:

77% (01:42) correct 23% (02:12) wrong based on 143 sessions

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Joined: 31 Oct 2013
Posts: 1429
Concentration: Accounting, Finance
GPA: 3.68
WE: Analyst (Accounting)
Re: What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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19 Dec 2018, 11:50
2
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

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

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

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

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.

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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, 01:25
2
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

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Joined: 04 Jan 2015
Posts: 3019
What is the remainder when 5^16 - 3^16 is divided by 8?  [#permalink]

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01 Jan 2019, 22:56

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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What is the remainder when 5^16 - 3^16 is divided by 8?   [#permalink] 01 Jan 2019, 22:56
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