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505-555 Level|   Exponents|   Remainders|      
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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 15 Sep 2020, 08:15
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If n is a positive integer, what is the remainder when \(3^{4n + 2} + 1\) is divided by 10?

A. 0
B. 1
C. 2
D. 3
E. 4
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A
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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 15 Sep 2020, 08:34
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If n is a positive integer, what is the remainder when 3^(4n+2) +1 is divided by 10?

A. 0
B. 1
C. 2
D. 3
E. 4

Explanation:
3^(4n+2)= (3^4n) * 3^2
=(...1)*9
=(...9)
I.e 3^(4n+2) expression will have digit 9 at its unit place.

Hence 3^(4n+2)+1 will have (9+1=10) zero at its unit place i.e it will be multiple of 10

Thus 3^(4n+2)+1 will be fully divisible by 10 and remainder=0

Hence A is the correct answer.

Regards,
Atul Pandey

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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 15 Sep 2020, 09:55
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IMO-A

3^4n+2 can be write as (3^2)^2n *3^2
3^2 is equal to 9 when 9 is divided by 10 remainder is -1
-1^2n is even number so it is equal to 1
1 is divided by 10 remainder is 1

= (3^2)^2n *3^2 /10+ 1/10........
= remainder........1*-1+1
= 0

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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 15 Sep 2020, 10:35
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Asked: If n is a positive integer, what is the remainder when \(3^{4n + 2} + 1\) is divided by 10?

3^4 = 81 & 3^2 = 9

Last digit of 3^{4n} = 81^n = 1
Last digit of 3^{4n+2} = 9*1 = 9
Last digit of 3^{4n + 2} + 1 = 0

IMO A
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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 17 Dec 2020, 04:32
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Whether n= 1 or 2 or any other number, in terms of cyclicity it will give us the same unit digit : 9
let's set n=1, we have 3^6+1. 3^6 unit's digit is 9
9+1= 10 and 10/10=1 with 0 remainder

Answer A)
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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 18 Jun 2023, 03:42
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\(3^{4n+2}=3^{2(n+1)}\)
This tells us that the digits of the 3 will be either 9 or 1. Notice that 2(n+1) can never be a multiple of 4. Clearly then the units digit must be 9, which would leave a remainder of 9/10, which combined with the remainder of 1/10 gives us a remainder of 0.
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If n is a positive integer, what is the remainder when 3^(4n + 2) + 1 26 Jan 2025, 22:49
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We can also use the concept of cyclicity here.
3 has a cyclicity of 4, as the unit's digit of 3 repeats after every 4 consecutive powers.
For example, 3^1 = 3, now when the power 1 is divided by cyclicity 4, we get the remainder as 1, so for any power of 3, when divided by 4, if it leaves the remainder as 1, then it's unit's digit will be same as 3^1
The same applies to 3^2, 3^3, and 3^4 (For 3^5, the unit's digit is again 3, it repeats).

Now here, since 4n+2 when divided by 4, will always leave the remainder as 2, 3^4n+2 will always have 9 as its unit's digit (Same as 3^2 which has 9 as its unit's digit).
9 + 1 gives 10, which has 0 as its unit's digit, it will always be divisible by 10.

I hope this helps!
All the best!
Bunuel
If n is a positive integer, what is the remainder when \(3^{4n + 2} + 1\) is divided by 10?

A. 0
B. 1
C. 2
D. 3
E. 4
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