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For what value of 'n' will the remainder of 351^n and 352^n be the sam

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Bunuel
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For what value of 'n' will the remainder of 351^n and 352^n be the sam [#permalink] New post   23 Jun 2020, 07:39
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For which of the following values of n will the remainder of 351^n and 352^n be the same when divided by 7?

I. 3
II. 4
III. 6

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

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C
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Re: For what value of 'n' will the remainder of 351^n and 352^n be the sam [#permalink] New post   23 Jun 2020, 08:07
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To find : for what value of n the remainder of 351^n/7 And 352^n/7 is N

Solution: (350+1)^n/7 = 1^n/7
=> (350+2)^n/7 = 2^n/7

I. 3
=> 1^3 mod 7 = 1 as remainder
=> 2^3 mod 7 = 8 mod 7 = 1 as remainder. (Matching)

II. 4
=> 1^4 mod 7 = 1 as remainder
=> 2^4 mod 7 = 16 mod 7 = 2 as remainder. (Not Matching)

III. 6
=> 1^6 mod 7 = 1 as remainder
=> 2^6 mod 7 = 64 mod 7 = 1 as remainder. (Matching)

Only statement 1&3 are matching.

IMO C

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For what value of 'n' will the remainder of 351^n and 352^n be the sam [#permalink] New post   30 Jul 2020, 03:09
Note that there is an inheritent pattern. Do solve such questions untimed, I usually look at the same principle tested with easier numbers.
Lets pick a number that gives a remainder of 2 when being divided by 7 -> 2

2 : 7-> R:2 let's replicate the pattern by raising the dividend by 2
4 : 7 -> R:4 first hint: the remainder was raised by 2 as well
8 : 7 -> R:1 if we would again raise the initial remainder by 3 (2^3) we would receive 8 as the remainder. As we are dividing by 7 we know that the remainder can never be greater or equal than the divisor, thus 8/7 -> R:1
16 : 7 -> R:2 at this point we should realize the patter 2-4-1

Same is applicable with 1 just that the remainder will always remain at one. We are thus looking for N such that it will result in a remainder of 1 for both expression. Remember that every 3rd remainder in the pattern was one. Thus we can safely assume that if n = 3 -> R:1 and if N = 6 -> R:1
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Re: For what value of 'n' will the remainder of 351^n and 352^n be the sam [#permalink] New post   04 Jul 2023, 22:14
Asked: For which of the following values of n will the remainder of 351^n and 352^n be the same when divided by 7?

The remainder when 351ˆn is divided by 7
= The remainder when (350+1)ˆn is divided by 7
= The remainder when (7*50+1)ˆn is divided by 7
=1

The remainder when 352ˆn is divided by 7
= The remainder when (350+2)ˆn is divided by 7
= The remainder when (7*50+2)ˆn is divided by 7
= The remainder when 2ˆn is divided by 7

I. 3
The remainder when 2ˆ3=8 is divided by 7 =1
The remainder when 351ˆ3 is divided by 7 = The remainder when 352ˆ3 is divided by 7 =1
II. 4
The remainder when 2ˆ4=16 is divided by 7 =2
The remainder when 351ˆ4 is divided by 7=1 <> The remainder when 352ˆ4 is divided by 7=2
III. 6
The remainder when 2ˆ6=64=7*9+1 is divided by 7 =1
The remainder when 351ˆ6 is divided by 7 = The remainder when 352ˆ6 is divided by 7 =1

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

IMO C
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Re: For what value of 'n' will the remainder of 351^n and 352^n be the sam [#permalink]
04 Jul 2023, 22:14
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