Find all three-digit even numbers N such that 693xN is a perfect square, that is, 693 x N = k2 where k is an interger.
Use the fact that any two perfect squares multiplied will produce another perfect square.
e.g. 4x9 = 36
693 = 9x7x11 so we would need multipliers of 7 and 11 to yield a perfect square
so 693x7x11 = 9x49x121
so N could be 77, but you wanted N to be even.
any number multiplied by an even will yield an even.
so N = 77x(any even perfect square)
e.g.
N = 77x4 or
N = 77x16 etc
308, yes exactly, I see.
Thank-you very much.
can you plz explain
693 = 9x7x11 so we would need multipliers of 7 and 11 to yield a perfect square
so 693x7x11 = 9x49x121
Thanks
To find the three-digit even numbers N such that 693xN is a perfect square, we can approach it step by step:
Step 1: Write the number 693 in its prime factorization form.
693 = 3 x 3 x 7 x 11
Step 2: Analyze the prime factors of 693 to determine the factors required for N to make 693xN a perfect square.
- The prime factor 7 needs to be squared because it appears only once in the factorization.
- The prime factor 11 needs to be squared because it appears only once in the factorization.
Step 3: Determine the remaining prime factors that should exist in the factorization of N to complete the perfect square.
- Since N is a three-digit even number, it must have three prime factors.
- We already identified two of the factors as 7 and 11 that need to be squared.
Step 4: Determine all possible combinations of prime factors for the third factor.
- Since N is a three-digit even number, the third factor must be a two-digit prime number.
- Candidates for the third factor are 2, 3, 5, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Step 5: Calculate N for each combination of prime factors to check if 693xN is a perfect square.
- Multiply the square of each prime factor together with 7 and 11, and check if the result is a perfect square.
- For example, if we take 2 as the third factor: N = 7^2 x 11^2 x 2 = 3388. Check if 693 x 3388 is a perfect square (e.g., 693 x 3388 = k^2).
- Perform this calculation for all possible combinations of prime factors.
Step 6: Identify the three-digit even numbers N that result in 693xN being a perfect square.
- If the product 693 x N is a perfect square, note down the corresponding value of N.
- Repeat this step for all the combinations of prime factors.
By following these steps, you should be able to determine the three-digit even numbers N such that 693xN is a perfect square.