1. ## 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

2. 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
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

3. 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?

4. 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..

5. 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

6. 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