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Bunuel
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What is the remainder when 63^26 is divided by 16? 05 Dec 2022, 07:00
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A
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35% (medium)

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65% (01:10) correct 35% (01:50) wrong based on 195 sessions

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What is the remainder when 63^26 is divided by 16?

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B. 3
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A
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What is the remainder when 63^26 is divided by 16? 06 Dec 2022, 20:15
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Bunuel
What is the remainder when 63^26 is divided by 16?

A. 1
B. 3
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D. 8
E. 10
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\(63^{26}=(64-1)^{26}=(16*4-1)^{26}\)
Thus, the remainder will be \((-1)^{26}\) or 1.

A
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What is the remainder when 63^26 is divided by 16? 07 Dec 2022, 01:35
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Bunuel
What is the remainder when 63^26 is divided by 16?

A. 1
B. 3
C. 5
D. 8
E. 10
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Solution:

  • We are asked the value of \((\frac{63^{26}}{16})_R\)
  • Ignoring the power 26, we have \((\frac{63}{16})_R=15\)
  • So, \((\frac{63^{26}}{16})_R\) will be equivalent to \((\frac{15^{26}}{16})_R\)
  • Now we know \((\frac{15^2}{16})_R=1\). So, \((\frac{15^{26}}{16})_R=(\frac{15^{2\times 13}}{16})_R=1\)

Hence the right answer is Option A
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What is the remainder when 63^26 is divided by 16? 02 Mar 2024, 11:13
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↧↧↧ Detailed Video Solution to the Problem ↧↧↧



What is the remainder obtained when \(63^{26}\) is divided by 16?

To solve this problem we will be using a concept called as Binomial Theorem
Learn more about Binomial Theorem in this video.

Now, we need to break 63 into two number
  • One number should be a multiple of 16 and should be close to 63 (i.e. 64)
  • Other number should be a small number to make the sum or difference as 63 (i.e. -1)
=> Remainder of \(63^{26}\) by 16 = Remainder of \((64-1)^{26}\) by 16

"The reason we are doing this is because when we open \((64-1)^{26}\) this using Binomial Theorem then we will get all the terms except one term as a multiple of 64 (which also makes them a multiple of 16."
=> Remainder of all the terms by 16, except one term will be 0

Let's open \((64-1)^{26}\) using Binomial Theorem to understand this

 \((64-1)^{26}\) = \(26C0 * 64^{26} * (-1)^0 + 26C1 * 64^{25} * (-1)^1 + .... + 26C25* 64^{1} * (-1)^{25} + 26C26* 64^{0} * (-1)^{26}\)

=> All terms except the last term are multiples of 64 => Their remainder by 16 will be 0

=> Our problem is reduced to what is the remainder when \(26C26* 64^{0} * (-1)^{26}\) is divided by 16

\(26C26* 64^{0} * (-1)^{26}\) = 1 * 1 * 1 = 1
=> Reminder of 1 by 16 = 1

So, Answer will be A
Hope it helps!

MASTER Remainders with 2, 3, 5, 9, 10 and Binomial Theorem

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What is the remainder when 63^26 is divided by 16? 28 Mar 2025, 23:47
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