What is the remainder when 4^23 is divided by 7?

Question

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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sujoykrdatta · Answer

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

sujoykrdatta · Answer

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

ManifestDreamMBA · Answer

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

LisaMMMM · Answer

Why are R1 and R2 multiplied in the end and not added?

Thanks

Krunaal · Answer

(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

smbb92 · Answer

Correction for fellow mates who are struggling at the first glance.

its not 43, its 4^3
its not 42, its 4^2

bumpbot · Answer

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What is the remainder when 4^23 is divided by 7?

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What is the remainder when 4^23 is divided by 7?

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