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Grace H.

asked • 09/15/14

Even multiple squares

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!

3 Answers By Expert Tutors

By:

Arthur D.

tutor
Here's a slightly different approach...
we want 14a to be a perfect square
14=2*7, therefore a=2*7
14*14=196
14=2*7, so we need a=2*2*2*7*7*7 which gives us an even number of 2s and 7s
a=2*2*2*7*7*7=2744
14*2744=38,416, or 196*196=38,416
196*196*196=7,529,536 which is too large
 
 
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09/15/14

Arthur D.

tutor
Here's another approach similar to Phillip R.'s first solution.
14a=n2, a and n are integers
14a=2*7*a, so a must be 2*7 to give us an even number of 2s and 7s
14*14=196
14a=2*7*a so a must be 2*2*2*7*7*7=2744
14*2744=38,416,or you could multiply 196*196
6 2s and 6 7s multiplied together is too large
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09/15/14

Phillip R. answered • 09/15/14

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Phillip R.

I forgot even times odd is even so odd squares are also good. Yes there are 36 + 35 = 71 numbers
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09/15/14

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