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655-705 (Hard)|   Exponents|   Remainders|      
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Bunuel
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What is the remainder when 4^23 is divided by 7? 24 Apr 2025, 00:30
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C
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65% (hard)

Question Stats:

58% (01:38) correct 42% (01:33) wrong based on 329 sessions

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What is the remainder when \(4^{23}\) is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

­
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C
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What is the remainder when 4^23 is divided by 7? 24 Apr 2025, 00:35
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Bunuel
What is the remainder when \(4^{23}\) is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

­
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We know 43 = 64, which leaves rem 1 when divided by 7

Thus: 4^23 = (43)7 × 42
(43)7 leaves rem 17 = 1 when divided by 7 and 42 = 16 leaves rem 2 when divided by 7
Thus, required rem = 1 × 2 = 2

Ans C
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What is the remainder when 4^23 is divided by 7? 24 Apr 2025, 00:37
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Bunuel
What is the remainder when \(4^{23}\) is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

­
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Method 2:

Using pattern:
4÷7 rem = 4
4^2÷7 rem = 2
4^3÷7 rem = 1
4^4÷7 rem = 4
And the cycle repeats. This is a cycle of 3 places.
Thus, 23÷3 leaves rem 2
Thus, we need the second place of the cycle => rem 2

Ans C
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What is the remainder when 4^23 is divided by 7? 25 Apr 2025, 11:14
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\(4^{23}\)
\((4^{3})^7 * 4^2\)
\((63+1)^7 * 4^2\)

63 is divisible by 7, (63+1)^7 will leave a remainder of 1
4^2=16 = 14+2, will leave a remainder of 2
R1 * R2 = R2
Bunuel
What is the remainder when \(4^{23}\) is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

­
Show more
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What is the remainder when 4^23 is divided by 7? 27 Apr 2025, 05:45
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Why are R1 and R2 multiplied in the end and not added?

Thanks
ManifestDreamMBA
\(4^{23}\)
\((4^{3})^7 * 4^2\)
\((63+1)^7 * 4^2\)

63 is divisible by 7, (63+1)^7 will leave a remainder of 1
4^2=16 = 14+2, will leave a remainder of 2
R1 * R2 = R2
Bunuel
What is the remainder when \(4^{23}\) is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

­
Show more
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What is the remainder when 4^23 is divided by 7? 27 Apr 2025, 14:53
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LisaMMMM
Why are R1 and R2 multiplied in the end and not added?

Thanks
ManifestDreamMBA
\(4^{23}\)
\((4^{3})^7 * 4^2\)
\((63+1)^7 * 4^2\)

63 is divisible by 7, (63+1)^7 will leave a remainder of 1
4^2=16 = 14+2, will leave a remainder of 2
R1 * R2 = R2
Show more

\((63+1)^7 * 4^2\) => R1 * R2

You can also approach this as follows:

\(4^{23} = 2^{46}\)

=> \(2^{45} * 2\)
=> \(8^{15} * 2\)

/ by 7, Remainder from \(8^{15}\) is 1 and from 2 is 2

=> 1*2 = 2
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What is the remainder when 4^23 is divided by 7? 18 Jun 2025, 18:56
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Correction for fellow mates who are struggling at the first glance.

its not 43, its 4^3
its not 42, its 4^2
sujoykrdatta
Bunuel
What is the remainder when \(4^{23}\) is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

­


We know 43 = 64, which leaves rem 1 when divided by 7

Thus: 4^23 = (43)7 × 42
(43)7 leaves rem 17 = 1 when divided by 7 and 42 = 16 leaves rem 2 when divided by 7
Thus, required rem = 1 × 2 = 2

Ans C
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