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# For prime number y, y>3. Which of the following could be the remainder

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For prime number y, y>3. Which of the following could be the remainder [#permalink]
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Bunuel wrote:
For prime number y, y>3. Which of the following could be the remainder when y^3 is divided by 12?

A. 2
B. 3
C. 4
D. 5
E. 6

Let’s test the first prime number greater than 3, which is 5.

5^3 = 125

125/12 = 10 remainder 5

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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
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Bunuel wrote:
For prime number y, y>3. Which of the following could be the remainder when y^3 is divided by 12?

A. 2
B. 3
C. 4
D. 5
E. 6

Prime number greater than 3 can be expressed as $$6k + 1$$ or $$6k -1$$.

$$\frac{(6k ± 1)^3}{12} = \frac{6^3k^3 ± 3*6^2k^2 + 3*6k ± 1}{12} = \frac{0 + 0 + 18k ± 1}{12} = \frac{6k ± 1}{12}$$

Let’s check first case $$6k + 1$$

k=1 (7/12) rem=7
k=2 (13/12) rem =1
k=3 (19/12) rem = 7
k =4 (25/12) rem = 1

we have cycle of remainders $$7$$ and $$1$$.

Next $$6k – 1$$

k=1 (5/12) rem=5
k=2 (11/12) rem =11
k=3 (17/12) rem = 5
k =4 (23/12) rem = 11

again cycle of only two remainders $$5$$ and $$11$$.

The only remainders after division by 12 cube of prime number $$y > 3$$ can have are $$1$$, $$5$$, $$7$$ and $$11$$.

Checking our options.

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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
vitaliyGMAT wrote:
Bunuel wrote:
For prime number y, y>3. Which of the following could be the remainder when y^3 is divided by 12?

A. 2
B. 3
C. 4
D. 5
E. 6

Prime number greater than 3 can be expressed as $$6k + 1$$ or $$6k -1$$.

$$\frac{(6k ± 1)^3}{12} = \frac{6^3k^3 ± 3*6^2k^2 + 3*6k ± 1}{12} = \frac{0 + 0 + 18k ± 1}{12} = \frac{6k ± 1}{12}$$

Let’s check first case $$6k + 1$$

k=1 (7/12) rem=7
k=2 (13/12) rem =1
k=3 (19/12) rem = 7
k =4 (25/12) rem = 1

we have cycle of remainders $$7$$ and $$1$$.

Next $$6k – 1$$

k=1 (5/12) rem=5
k=2 (11/12) rem =11
k=3 (17/12) rem = 5
k =4 (23/12) rem = 11

again cycle of only two remainders $$5$$ and $$11$$.

The only remainders after division by 12 cube of prime number $$y > 3$$ can have are $$1$$, $$5$$, $$7$$ and $$11$$.

Checking our options.

I have a doubt here please....prime numbers graeter than 3 can be expressed as 6k+1..putting k=4 we get 25, which is not a prime num,ber....am I missing something here?
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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
saurabhsavant wrote:
vitaliyGMAT wrote:
Bunuel wrote:
For prime number y, y>3. Which of the following could be the remainder when y^3 is divided by 12?

A. 2
B. 3
C. 4
D. 5
E. 6

Prime number greater than 3 can be expressed as $$6k + 1$$ or $$6k -1$$.

$$\frac{(6k ± 1)^3}{12} = \frac{6^3k^3 ± 3*6^2k^2 + 3*6k ± 1}{12} = \frac{0 + 0 + 18k ± 1}{12} = \frac{6k ± 1}{12}$$

Let’s check first case $$6k + 1$$

k=1 (7/12) rem=7
k=2 (13/12) rem =1
k=3 (19/12) rem = 7
k =4 (25/12) rem = 1

we have cycle of remainders $$7$$ and $$1$$.

Next $$6k – 1$$

k=1 (5/12) rem=5
k=2 (11/12) rem =11
k=3 (17/12) rem = 5
k =4 (23/12) rem = 11

again cycle of only two remainders $$5$$ and $$11$$.

The only remainders after division by 12 cube of prime number $$y > 3$$ can have are $$1$$, $$5$$, $$7$$ and $$11$$.

Checking our options.

I have a doubt here please....prime numbers graeter than 3 can be expressed as 6k+1..putting k=4 we get 25, which is not a prime num,ber....am I missing something here?

Any prime number $$p>3$$ when divided by 6 can only give remainder of 1 or 5 (remainder can not be 2 or 4 as in this case $$p$$ would be even and remainder can not be 3 as in this case $$p$$ would be divisible by 3).

So any prime number $$p>3$$ could be expressed as $$p=6n+1$$ or$$p=6n+5$$ or $$p=6n-1$$, where n is an integer >1.

But:
Not all number which yield a remainder of 1 or 5 upon division by 6 are prime, so vise-versa of above property is not correct. For example 25 yields a remainder of 1 upon division be 6 and it's not a prime number.

Hope it's clear.
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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
prime number 5 , 5^3 /12 = 25*5 /12 => 1*5/12 ( since 25/12 will give rem as 1) => 5 is rem

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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
We can check it by plugging prime numbers greater than 3.

For Ex. Take prime number 5. Y^3 = 125 and divide by 12 then remainder become 5. Hence, Answer is D

Hope it helps !
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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
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Re: For prime number y, y>3. Which of the following could be the remainder [#permalink]
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