Problem of the Week 9

Is 14,798,678,562 a perfect square?  Is 15,763,530,163,289 a perfect square?  Examine both the units digits and the digital roots* of perfect squares to help determine whether or not a given number is a perfect square.  Explain your answers.

*The digital root of 579 is 3 because 5 + 7 + 9 = 21 and 2 + 1 = 3.  The digital root of 183,520 is 1 because 1 + 8 + 3 + 5 + 2 + 0 + 19 and 1 + 9 = 10 and 1 + 0 = 1.

 

Alternate Problem 9

Suppose a = (((22) 2) 2)and b = 216.  Find the value of b/a in the form 2p by replacing p by the proper integer.

 

Extension Problem 9

Write as many as possible of the first 21 whole numbers as a difference  of two squares.  For example, 02 = 02 – 02 and 8 = 32 – 12.  Which of the whole numbers fail to be expressible in the form of two squares?

Make a generalization about the results.