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When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5

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Bunuel
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When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5 [#permalink] New post   18 Nov 2023, 06:45
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When n is divided by 4 the remainder is 3. If \(3^n+2\) is divided by 5, what is the remainder?

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B. 2
C. 3
D. 4
E. 0
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sachin_27
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Re: When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5 [#permalink] New post   19 Nov 2023, 04:31
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n/4 = remainder is 3.
taking n = 7, therefore remainder is 3.

3^n+2/5 = 3^7+2/5 = 2189/5 = remainder is 4.
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Re: When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5 [#permalink] New post   22 Nov 2023, 10:40
sachin_27 wrote:
n/4 = remainder is 3.
taking n = 7, therefore remainder is 3.

3^n+2/5 = 3^7+2/5 = 2189/5 = remainder is 4.


How can u solve 3^7 without using a calculator?
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Re: When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5 [#permalink] New post   22 Nov 2023, 20:47
well, it's easier if you few digits with their powers up till 10 and first 10 factorials and their values.
or let me tell you a simple way
3^7 = 3.3.3.3.3.3.3
so rather than actually multiplying 3 with itself 7 times, you can club it up = 9.9.9.3 = 729*3
i hope this helps.
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Re: When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5 [#permalink] New post   24 Nov 2023, 16:40
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n = 4q + 3
possible values of n are 3,7,11,15,19...

The cyclicity of powers of 3 is: 3,9,7,1

By looking at all the possible values of n, you can quickly see that they will always end up being in the third place of the cyclicity of powers of 3 (as you increase every possible value of n by 4 every time)

Therefore, the units digit will always be a 7

therefore the remainder of 3^n + 2 will always be 9/5 = 4
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Re: When n is divided by 4 the remainder is 3. If 3^n + 2 is divided by 5 [#permalink]
24 Nov 2023, 16:40
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