Make It a Perfect Square

Number S can be written as:

S = 15 + a + 18 + b + 30,

with all the addends in ascending order and different from each other.

What values can a and b take knowing that S is a perfect square?

 

Answer

All the addends are in ascending order and different from each other, therefore

15 < a < 18, and                   (1)

18 < b < 30.                            (2)

Then 33 < a + b < 48.          (3)

Also, note that we can rewrite S as

S = a + b + 63.                        (4)

Adding 63 to both sides of inequality (3), we get:

96 < a + b + 63 < 111, or

96 < S < 111.

S has to be a perfect square, and the only perfect square between 96 and 111 is 100.

From (4) we know that a + b = S – 63, therefore

a + b = 100 – 63 = 37.

From (1) we know that the values a can take are 16 and 17.

For a = 16, b = 37 – 16 = 21.

For a = 17, b = 37 – 17 = 20.

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