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

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

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New post 19 Dec 2018, 11:18
1
10
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A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

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

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

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New post 19 Dec 2018, 11:50
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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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\(5^{16} - 3^{16}\)

\(5^8*2 - 3^8*2\)

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

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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New post 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

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

Image

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

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New post 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.

Answer: 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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