# yr 8 homework

• Jul 25th 2008, 06:57 PM
mathswonderer
yr 8 homework
Find all the three-digit even numbers N such that 693 x N is a perfect square, that is 693 x N = k squared whre k is an integer.

I know that 308 is oneof them but how do we no that it is the only one? are there more?

I think I bascaly kno but I just want to be sure, please explain carefully(Crying)
• Jul 25th 2008, 07:33 PM
kalagota
Quote:

Originally Posted by mathswonderer
Find all the three-digit even numbers N such that 693 x N is a perfect square, that is 693 x N = k squared whre k is an integer.

I know that 308 is oneof them but how do we no that it is the only one? are there more?

I think I bascaly kno but I just want to be sure, please explain carefully(Crying)

first, get the factorization of 693:

693 = (3^2)(7)(11)

now, start finding \$\displaystyle N\$ such that it has factors 7 and 11 whose powers are odd, and other integer factors whose powers are even.

for example: N= (7)(11)(2^2) = 308 and you are correct..

N= (7)(11)(3^2) = 693
N= (7)(11)(4^2) = ?.. just make sure that \$\displaystyle N\$ is still a 3 digit number..

EDIT: basically, i chose 1 as the powers of 7 and 11, since it it were 3 or higher odd powers, then it will give you not a three digit number.. EX: (7^3)(11) = 3773
• Jul 25th 2008, 07:50 PM
mathswonderer
I understand that but how do we know that 308 is the only one, how can I simtimatically show that it is the onnly one?
• Jul 25th 2008, 07:57 PM
kalagota
308 is not the only one..

note that 693 is also a 3-digit number, and if you multiply it to itself, it will give you a perfect number,,

have you solve the last N? is it a 3-digit number? if it is, then you got another number other that 308 and 693.. and if not, then there are no other numbers..
• Jul 26th 2008, 11:59 PM
mathswonderer
SO would this working be sufficient?
1. 7 x 11 x 2= 154
693 x 154=106 722
Sqrt=326.683332

2. 7 x 11 x 22=308
693 x 308 = 213 444
Sqrt=462

3. 7 x 11 x 23=616
693 x 616=426 888
Sqrt=653.36666

4. 7 x 11 x 24=1232
693 x 1232= 853 776
Sqrt=924
• Jul 28th 2008, 04:23 AM
kalagota
Quote:

Originally Posted by mathswonderer
SO would this working be sufficient?
1. 7 x 11 x 2= 154
693 x 154=106 722
Sqrt=326.683332

2. 7 x 11 x 2x2=308
693 x 308 = 213 444
Sqrt=462

3. 7 x 11 x 2x3=616
693 x 616=426 888
Sqrt=653.36666

4. 7 x 11 x 2x4=1232
693 x 1232= 853 776
Sqrt=924

1. NO! since the square root is not an integer..
2. yes!
3. NO! same reason as 1.
4. NO! since 1232 is not a 3-digit number..

besides, for number 3: 7 x 11 x 2 x 3 = 462
for number 4: 7 x 11 x 2 x 4 = 616