Why?

In mathematics one of the most important questions is why. Let us consider a problem:

Problem. A number has three hundred ones and three hundred zeroes. Can it be a square?

The solution goes like this. Consider divisibility of this number by 9. The sum of the digits is 300. That means the number is divisible by 3, but not by 9. Therefore, it can’t be a square.

Why do we consider divisibility by 9? The divisibility by 9 is a very powerful tool, but why was it the first thing that came to my mind? The divisibility by 9 doesn’t depend on the order of the digits. Whenever I see a problem where the question talks about digits that can be in any order, the first tool to use is the divisibility by 9.

The why question, is very important in mathematics. But it is also very important in life. It took me many years to start asking why people did this or that. I remember my mom was visiting me in the US. Every time I came back from work, she complained that she was tired. Why? Because she did the laundry in the bath tub. She wouldn’t use my washing machine, because she didn’t have such a thing in Russia. I promised her that I’d do the laundry myself when there was a sufficient pile. However, she insisted that the dirty clothes annoyed her. I would point that my water bill went up. And so on.

We argued like this every day. We were both frustrated. Then I asked myself why. Why does she do the laundry? The answer was there. She wanted to be helpful. I calmed down and stopped arguing with her. I sucked it up and paid the water bills. Her time with me turned into the most harmonious visit we ever had. Unfortunately, it was the last.

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1. David Reynolds:

The binary number
101010101010101010101010101010101010101010101010101010101010
101010101010101010101010101010101010101010101010101010101010
101010101010101010101010101010101010101010101010101010101010
101010101010101010101010101010101010101010101010101010101010
101010101010101010101010101010101010101010101010101010111011
011110011111001111010110001001110101111110010010001101001001
101100101010101010111111100100111011111011011000110010101110
001000000010010000011110101110011101111011001000010000101000
100001101100000101010010110110010110110100110001101001000110
111010100100101100011000110001011111011010000011011001000000
has 300 zeroes and 300 ones and is equal to
1663232909903880130353791784875948825457990284989189431854443510921448232557386925018638552
squared.