The Magical Number 37

The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.

John William Navin SULLIVAN

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The natural number 37 lying between 36 and 38 plays an important role in mathematics. In this blog post let us discuss it. Consider any three-digit number with all digits equal. For example, 111, 222, etc. These numbers are all divisible by 37. Moreover, the answer is just given by adding all the digits. Let’s see:

While dividing 111 by 37, we get 3 as the answer, which can be written as 1 + 1 + 1 and this is the sum of digits in the given number.

Next, consider the number 222. While dividing by 37, we get 6 as the answer. Let’s see:

As before, while dividing 222 by 37, we get 6 as the answer, which can be written as 2 + 2 + 2 and this is the sum of digits in the given number.

Following the same rule as above, we have

Few more examples are given below.

How is this happening? What’s the mathematical secret behind this phenomenon? Let’s analyze.

Any threedigit number like aaa can be written in expanded form as 100a + 10a + a, which is equal to 111a. Certainly, 111 is always divisible by 37 and we can write 111a = 37 × 3a. When a = 1, we get the number as 111; for a = 2, the number is 222; for a = 3, the number is 333 and so on.

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