Even multiple squares

Question

Could someone explain how to solve this problem please?

 

How many integers exist that are even multiples of 7 between 1 and 1,000,000 and are also squares? 

 

Thanks!

Answer 1

14*14=196 has to be the first

1,000,000/196≈5102

√5102≈71

4*196=784

9*196=1764

16*196=3136

25*196=4900

36*196=7056

49*196=9604

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4761*196=933,156

4900*196=960,400

5041*196=988,036

there should be 71 numbers that are even multiples of 7 and that are perfect squares

 

Answer 2

Even multiples of 7 are always multiples of  (2*7) or 14.

For example, 14, 28, 42, 56, ...

Since the square of each even multiple is less than or equal to 1,000,000, then the even multiple of 7 must be less than or equal to (√1,000,000) or 1,000.

Dividing to find the number of specified integers we get

1,000/14 = 71.428 which truncates to 71 integers.

 

You can also build a spreadsheet to actually find all of these integers and count them! The first is 196 and the last is 988,036.

 

I would like to see another approach to this interesting problem from another tutor!

Answer 3

even multiples of 7 start at 14 and the first perfect square is 14². The subsequent perfect squares will be 196(4n²) n = 1,2,3,...

We stop when 4n² is less than or equal to 1,000,000/196 = 5102 or n² is less than or equal to 1276. The square root of 1276 is 35.72. So there are 35 + 1 = 36 numbers between 1 and 1,000,000 that are perfect squares and also even multiples of 7.