Suddenly I noticed a common property of any number while solving a problem like what is the sum of digits of the product of 61s 3 that means 33333.........3333 and 62s 6 means 666......6666.For 3 and 6, as 6 is twice of 3, the sum of digits of their product follow a specific rule.

After solving the problem , i thought that , Does every number contains this property ? So I tried to find the sequence for 2 and 4 also. As 4 and 8 are twice of 2 and 4 , so i considered the products of any certain number of 2 suppose **n** , and **n+1** numbers of 4. Then i took product of the numbers and then calculated the sum of digits of the number combined with 2 and 4 like **22444** and then calculate the sum of digits of the product of two numbers 22 and 444 . Then i calculated **Sum Of Digits(Products)- Sum Of Digits(Numbers)** . And i saw that the result follow a rule . Just see the image bellow to understand what I've done :

Please notice the series carefully. If you subtract 1st term from 2nd term then you'll get 8. By the same process the difference of every 2 term till 10th term are 8, 6, 4, 2, 0, -2,- 4, 12 and 1. Then this repeats again and again till infinity. So , It's clear that there is a strong logic behind this sequence.

Now for the number 4, I did same process again . And at first see the image :

Now notice that after 20th term the **Sum Of Digits(Numbers) - Sum Of Digits(Products)** follows a sequence. it is :
-7
-7
-7
-7
2
-7
-7
-7
-7
2
-7
-7
-7
-7
2
-7
-7
-7
-7
2
And the mod of **Sum Of Digits(Numbers)% Sum Of Digits(Products)** follow a specific rule after 20th term :
64
66
68
70
2
65
67
69
71
2
66
68
70
72
2
67
69
71
73
2

You can try it for any other numbers too. Thanks for reading this. You can download my Microsoft Excel files from my website MathMad.

## Comments

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TopNewestIf a hemispherical bowl is 4 cm long. It is filled upto 3 cm. What volume of bowl is filled? – Himanshi Yadav · 8 months, 2 weeks ago

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Can you explain why this pattern exists?

What is the math behind it? How would we find the sum of digits of this product? – Calvin Lin Staff · 8 months, 2 weeks ago

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– অরিন্দম পাল · 8 months, 2 weeks ago

I don't know why this types of pattern exists ? But i found different logic for different numbers. If you know , please explain in details. I want to know more about this. But i believe that patterns and numbers are everywhere. I worked for a thesis paper last year and their i also found a pattern . I don't know about any specific reason behind it.Log in to reply