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Let n be a nonnegative integer such that when 6n is divided by 75, the

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Afroditee
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Let n be a nonnegative integer such that when 6n is divided by 75, the [#permalink] New post   16 Nov 2022, 07:24
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Let n be a nonnegative integer such that when 6n is divided by 75, the remainder is 30. Which of the following is a list of all possible remainders when 7n is divided by 75?

A 5, 35

B 10, 55

C 35, 60

D 5, 30, 55

E 10, 35, 60
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E
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Re: Let n be a nonnegative integer such that when 6n is divided by 75, the [#permalink] New post   16 Nov 2022, 08:22
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Afroditee wrote:
Let n be a nonnegative integer such that when 6n is divided by 75, the remainder is 30. Which of the following is a list of all possible remainders when 7n is divided by 75?

A 5, 35

B 10, 55

C 35, 60

D 5, 30, 55

E 10, 35, 60



Given: \(6n=75k+30\)
The above equation tells us two things
k is an even number, say k=2x
n is a multiple of 5, say n=5y

\(7n=6n+n=(75k+30)+n=75k+30+n\)
Thus, when 7n is divided by 75, the remainder is 30+n

Possible cases of n from 6n=75*2x+30=150x+30
1) x=0, then 6n=30 or n=5.
R= 30+n = 30+5 =35
2) x=1, then 6n=180 or n=30.
R= 30+n = 30+30 =60
3) x=2, then 6n=330 or n=55.
R= 30+n = 30+55 = 85 = 75+10 or 10.
4) x=3, then 6n=480 or n=80.
R= 30+n = 30+80 = 110 = 75+35 or 35, which is a repetition

Thus different possible remainders are 10, 35 and 60.

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Let n be a nonnegative integer such that when 6n is divided by 75, the [#permalink] New post   06 Dec 2022, 23:34
bb Bunuel @chetan4u @veritasprep any other way to solve ?
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Re: Let n be a nonnegative integer such that when 6n is divided by 75, the [#permalink] New post   07 Dec 2022, 00:48
KarishmaB pls help
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Re: Let n be a nonnegative integer such that when 6n is divided by 75, the [#permalink] New post   07 Dec 2022, 20:55
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Afroditee wrote:
Let n be a nonnegative integer such that when 6n is divided by 75, the remainder is 30. Which of the following is a list of all possible remainders when 7n is divided by 75?

A 5, 35

B 10, 55

C 35, 60

D 5, 30, 55

E 10, 35, 60



When 6n is divided by 75, the remainder is 30.
So 6n must be of the form 75a + 30 i.e. some values that 6n can take are 30, 105, 180, 255, 330 etc. Only the alternate terms here are of the form 6n where n is an integer.
So n can be 5, 30, 55 ... etc. We see a pattern.

So 7n can be 35, 210, 385.. etc. Getting 3 such numbers is enough since the maximum number of remainders in an option is 3.
So the remainders when 7n is divided by 75 can be 35, 60 and 10.

Answer (E)
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Re: Let n be a nonnegative integer such that when 6n is divided by 75, the [#permalink]
07 Dec 2022, 20:55
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