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Sub 505 (Easy)|   Exponents|   Remainders|      
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Bunuel
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What is the remainder when 7^442 is divided by 10? 18 Nov 2023, 07:08
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E
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83% (00:50) correct 17% (01:27) wrong based on 160 sessions

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What is the remainder when \(7^{442}\) is divided by 10?

A. 1
B. 3
C. 5
D. 7
E. 9
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E
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What is the remainder when 7^442 is divided by 10? 18 Nov 2023, 15:56
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Using the cyclicity of powers, the powers of 7 will always have the following unit digits:
7,9,3,1
the power is 442, had it been 444 the unit digit would have been a 1 (since 444 is divisible by 4), therefore, the unit digit will be two steps before that, i.e. ---> 9

9/10 leaves a remainder of 9

Answer E
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What is the remainder when 7^442 is divided by 10? 18 Nov 2023, 16:19
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Using cyclicity of power:
7^1 =7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
(And the cycle continues)

442 is not divisible by 4, but 444 is divisible so at 7^4 the unit digit will be 1, since 444 is +2 greater than 442 (go back twice), 442 will be = 9 [E]

I hope this helps

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What is the remainder when 7^442 is divided by 10? 16 Sep 2024, 04:53
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What is the remainder when \(7^{442}\) is divided by 10

Theory: Remainder of a number by 10 is same as the unit's digit of the number

(Watch this Video to Learn How to find Remainders of Numbers by 10)

Using Above theory Remainder of \(7^{442}\) by 10 = unit's digit of \(7^{442}\)

Now to find the unit's digit of \(7^{442}\) , we need to find the pattern / cycle of unit's digit of power of 7 and then generalizing it.

Unit's digit of \(7^1\) = 7
Unit's digit of \(7^2\) = 9
Unit's digit of \(7^3\) = 3
Unit's digit of \(7^4\) = 1
Unit's digit of \(7^5\) = 7

So, unit's digit of power of 7 repeats after every \(4^{th}\) number.
=> We need to divided 442 by 4 and check what is the remainder
=> 442 divided by 4 gives 2 remainder

=> \(7^{442}\) will have the same unit's digit as \(7^2\)
=> Unit's digits of \(7^{442}\) = 9

So, Answer will be E
Hope it helps!

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

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