Question

Jenny multiplies the square root of her favorite positive integer by $\sqrt{2}$. Her product is an integer. a) Name three numbers that could be Jenny's favorite positive integer, and explain why each could possibly be Jenny's favorite integer. b) Suppose Jenny divides the square root of her favorite positive integer by sqrt(2). Does she have to get an integer? (Remember, when Jenny multiplies the square root of her favorite integer by sqrt(2), she gets an integer.)

Answer 1

you can copy and paste this one for the Aops writing problem.

WARNING: this is to help you with your homework writing problem for Aops and not for cheating on it.

PS. Edit it and make it your own

Explanation:

(a) Three numbers that could be Jenny's favorite number are;

2, 8, and 18

(b) Yes

Explanation:

(a) Let Jenny's favorite integer be X, we have;

√X × √2 = Y where Y is an integer

Therefore, the square root of Jenny's favorite number has a factor of √2 which gives the possible options as

2 with √2 being the square root

8 with √8 = 2·√2

18 with √18 being 3·√2

(b) By dividing each of the square root of the possible Jenny's favorite number, we have;

i) For the integer 2 we have;

√2/√2 = 1 which is an integer

ii) For the integer 8 we have;

√8/√2 = 2·√2/√2 = 2 which is an integer

ii) For the integer 18 we have;

√18/√2 = 3·√2/√2 = 3 which is an integer

Answer 2

Part a)
`2,50,18`

Part b) When Jenny divides the square root of her favorite positive integer by
`√(2)`, she gets an integer

Explanation:

Let

x-------> the favorite positive integer

Part a)

1) For
`x=2`

`√(2)*√(2)=√(4)=2` -----> the product is an integer

so

The number
`x=2` could be Jenny favorite positive integer

2) For
`x=50`

`√(50)*√(2)=√(100)=10` -----> the product is an integer

so

The number
`x=50` could be Jenny favorite positive integer

3) For
`x=18`

`√(18)*√(2)=√(36)=6` -----> the product is an integer

so

The number
`x=18` could be Jenny favorite positive integer

Part B)

1) For
`x=2`

`√(2)/√(2)=√(1)=1` -----> the result is an integer

2) For
`x=50`

`√(50)/√(2)=√(25)=5` -----> the result is an integer

3) For
`x=18`

`√(18)/√(2)=√(9)=3` -----> the result is an integer

Therefore

When Jenny divides the square root of her favorite positive integer by
`√(2)` , she gets an integer