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# If n is a prime number and n ≠ 3, which of the following

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If n is a prime number and n ≠ 3, which of the following [#permalink]

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03 Nov 2012, 00:49
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If n is a prime number and n ≠ 3, which of the following could be the remainder when 100! + n is divided by 3?

I. 0
II. 1
III. 2

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

Bunuel - I checked out reminders concepts in the math book compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html
but the below idea wasn't mentioned

You can add and subtract remainders directly, as long as you correct excess or negative remainders. "

if x leaves a remainder of 4 after division by 7, and y leaves a remainder of 2 after division by 7, then x +y leaves a remainder of 4 + 2 = 6 after division by 7.

Similarly 100! leaves a remainder 0 on division by 3 so we are only interested in the remainder when N divided by 3,which will be actual remainders of 100! + n is divided by 3

if n =1 remainder 1 so overall remainder is 1
if n = 2 remainder 2 so overall remainder is 2
if n = 6 remainder = 0 so overall remainder is 0

All 3 are possible right then why Princeton says D?

Cheers.
[Reveal] Spoiler: OA

Last edited by Bunuel on 03 Nov 2012, 00:55, edited 2 times in total.
Renamed the topic and edited the question.
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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03 Nov 2012, 00:59
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Expert's post
Jp27 wrote:
If n is a prime number and n ≠ 3, which of the following could be the remainder when 100! + n is divided by 3?

I. 0
II. 1
III. 2

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

Bunuel - I checked out reminders concepts in the math book compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html
but the below idea wasn't mentioned

You can add and subtract remainders directly, as long as you correct excess or negative remainders. "

if x leaves a remainder of 4 after division by 7, and y leaves a remainder of 2 after division by 7, then x +y leaves a remainder of 4 + 2 = 6 after division by 7.

Similarly 100! leaves a remainder 0 on division by 3 so we are only interested in the remainder when N divided by 3,which will be actual remainders of 100! + n is divided by 3

if n =1 remainder 1 so overall remainder is 1
if n = 2 remainder 2 so overall remainder is 2
if n = 6 remainder = 0 so overall remainder is 0

All 3 are possible right then why Princeton says D?

Cheers.

Notice that we are told that n is a prime number and n ≠ 3. Thus, n cannot be 1.

n also cannot be 6 or any other multiple of 3, thus the remainder cannot be 0.

It can be 1 for n=7 and 2 for n=5.

Hope it's clear.
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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03 Nov 2012, 01:14
Bunuel wrote:
Jp27 wrote:
If n is a prime number and n ≠ 3, which of the following could be the remainder when 100! + n is divided by 3?

I. 0
II. 1
III. 2

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

Bunuel - I checked out reminders concepts in the math book compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html
but the below idea wasn't mentioned

You can add and subtract remainders directly, as long as you correct excess or negative remainders. "

if x leaves a remainder of 4 after division by 7, and y leaves a remainder of 2 after division by 7, then x +y leaves a remainder of 4 + 2 = 6 after division by 7.

Similarly 100! leaves a remainder 0 on division by 3 so we are only interested in the remainder when N divided by 3,which will be actual remainders of 100! + n is divided by 3

if n =1 remainder 1 so overall remainder is 1
if n = 2 remainder 2 so overall remainder is 2
if n = 6 remainder = 0 so overall remainder is 0

All 3 are possible right then why Princeton says D?

Cheers.

Notice that we are told that n is a prime number and n ≠ 3. Thus, n cannot be 1.

n also cannot be 6 or any other multiple of 3, thus the remainder cannot be 0.

It can be 1 for n=2 and 2 for n=5.

Hope it's clear.

ohh my god. Ok thanks.
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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03 Nov 2012, 07:30
Jp27 wrote:
If n is a prime number and n ≠ 3, which of the following could be the remainder when 100! + n is divided by 3?

I. 0
II. 1
III. 2

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

Bunuel - I checked out reminders concepts in the math book compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html
but the below idea wasn't mentioned

You can add and subtract remainders directly, as long as you correct excess or negative remainders. "

if x leaves a remainder of 4 after division by 7, and y leaves a remainder of 2 after division by 7, then x +y leaves a remainder of 4 + 2 = 6 after division by 7.

Similarly 100! leaves a remainder 0 on division by 3 so we are only interested in the remainder when N divided by 3,which will be actual remainders of 100! + n is divided by 3

if n =1 remainder 1 so overall remainder is 1
if n = 2 remainder 2 so overall remainder is 2
if n = 6 remainder = 0 so overall remainder is 0

All 3 are possible right then why Princeton says D?

Cheers.

if a number n is prime,
and together with that does not equal 3
we can divide it on 3 whithout a remainder.

we can also check it with, for example 5! and list of primes such as 2!,5!,7!...

if n = 6 - 6 is not prime
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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03 Nov 2012, 17:30
Bunuel wrote:
Jp27 wrote:
If n is a prime number and n ≠ 3, which of the following could be the remainder when 100! + n is divided by 3?

I. 0
II. 1
III. 2

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

Bunuel - I checked out reminders concepts in the math book compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html
but the below idea wasn't mentioned

You can add and subtract remainders directly, as long as you correct excess or negative remainders. "

if x leaves a remainder of 4 after division by 7, and y leaves a remainder of 2 after division by 7, then x +y leaves a remainder of 4 + 2 = 6 after division by 7.

Similarly 100! leaves a remainder 0 on division by 3 so we are only interested in the remainder when N divided by 3,which will be actual remainders of 100! + n is divided by 3

if n =1 remainder 1 so overall remainder is 1
if n = 2 remainder 2 so overall remainder is 2
if n = 6 remainder = 0 so overall remainder is 0

All 3 are possible right then why Princeton says D?

Cheers.

Notice that we are told that n is a prime number and n ≠ 3. Thus, n cannot be 1.

n also cannot be 6 or any other multiple of 3, thus the remainder cannot be 0.

It can be 1 for n=2 and 2 for n=5.

Hope it's clear.

Hi,

A minor correction in your post:
for n=2, the remainder will be 2, not 1.
for n=7, remainder will be 1.

With respect,
CJ
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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26 Nov 2012, 10:22
Another way to look at it is:

100!+n where n ≠ 3, since 100! will be a factor or 3, so we just have to care about n.
Hence, if n=2 then remainder of 2/3 is 2.
for any value of n>3, and n being prime it can be written as (6k+1) or (6k-1).
Hence, factor (6k+1)/3 will give remainder as 1, and (6k-1) would leave remainder as 2.

Please correct me if I am wrong.

Regards,
Nityam
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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06 Apr 2013, 05:37
100! is divisible by 3. So we need to find out the remainder when the prime number n is divided by 3.

For n = 2 or 5, remainder is 2.
For n = 7, remainder is 1.

n cannot be 3 as specified and cannot be any other multiple of 3 as n is prime. So the remainder cannot be 0.

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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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08 Aug 2014, 08:13
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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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13 Aug 2014, 01:45
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$$\frac{100! + n}{3} = \frac{100!}{3} + \frac{n}{3}$$

$$\frac{100!}{3}$$ is a perfect division

$$\frac{n}{3}$$ may give remainder either 1 or 2 as n is prime

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Re: If n is a prime number and n ≠ 3, which of the following [#permalink]

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20 Nov 2015, 04:57
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Re: If n is a prime number and n ≠ 3, which of the following   [#permalink] 20 Nov 2015, 04:57
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