The Mysterious Number 22

Number 22 Everywhere?

Numbers never fail to surprise us. This post talks about one such amazing property of number 22.

Select any three-digit number with all digits different from one another. Write all possible two-digit numbers that can be formed from the three-digits selected earlier. Then divide their sum by the sum of the digits in the original three-digit number.

You’ll always get the same answer, 22. Isn’t this wonderful!

For example, take the three-digit number 786. The 2 digit-numbers which can be made using the digits 7, 8 and 6 are 78, 87, 76, 67, 86, 68. Hence sum = 78 + 87 + 76 + 67 + 86 + 68 = 462. Sum of digits of 786 = 7+8+6 = 21. Then 462/21 = 22

This will be true for any three-digit number with all digits different.

Is it actually Mysterious?

Not really! If we go deeper and try to analyze this unusual result using, we’ll be able to appreciate the logic of it.

The general representation of any three digit number with all digits different will be 100x+10y+z.  Now to find the sum of all the two-digit numbers taken from the three digits

= (10x+y)+ (10y+x)+(10x+z)+(10z+x)+(10y+z)+(10z+y)
= 20(x+y+z) + 2(x+y+z)

This when divide by the sum of the digits, (x+y+z), is 22. This shows the importance of algebra in explaining such simple yet interesting mathematical phenomenon.

Do you know any such interesting property of any number?

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