I hope the readers did not find the first article difficult to digest. The readers are advised not to hurry themselves. They are to go through these thoughts with absolute peace of mind, for the mind shall reveal its true potential one day for a good cause. No more lectures. Let us get back to mathematics. Have you noticed how a small seed holds the key to a big tree? Here is a question to prove the point-
To know the answer to this question, you will have to pay homage to the simplest of all rules- the digit-sum rule.
What is Digit Sum?
Given a number N1, all the digits of N1 are added to obtain a number N 2 . All the digits of N2 are added to obtain a number N3, and so on, till we obtain a single digit number N. This single digit number N is called the digit sum of the original number N1.
Example: What is the digit sum of 123456789?
Answer: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 --> 4 + 5 = 9. Hence, the digit sum of the number is 9.
Note: In finding the digit-Sum of a number we can ignore the digit 9 or the digits that add up to 9. For example, in finding the digit-sum of the number 246819, we can ignore the digits 2, 6, 1, and 9. Hence, the digit-sum of 246819 is = 4 + 8 = 12 = 1 + 2 = 3.
Digit-Sum Rule of Multiplication: The digit-sum of the product of two numbers is equal to the digit sum of the product of the digit sums of the two numbers!
Example: The product of 129 and 35 is 4515.
Digit sum of 129 = 3 and digit sum 35 = 8
Product of the digit sums = 3 × 8 = 24 --> Digit-sum = 6.
Digit-sum of 4515 is = 4 + 5 + 1 + 5 = 15 = 1 + 5 = 6.
Digit-sum of the product of the digit sums = digit sum of 24 = 6
--> Digit sum of the product (4515) = Digit-sum of the product of the digit sums (24) = 6
Applications of Digit-Sum
- Rapid checking of calculations while multiplying numbers
Suppose a student is trying to find the product 316 × 234 × 356, and he obtains the number 26525064.
A quick check will show that the digit-sum of the product is 3. The digit-sums of the individual numbers (316, 234 and 356) are 1, 9, and 5. The digit-sum of the product of the digit sum is 1 × 9 × 5 = 45 = 4 + 5 = 9.
--> the digit-sum of the product of the digit-sums (9) is NOT equal to digit-sum of the 26525064 (3)
Hence, the answer obtained by multiplication is not correct.
Note: Although the answer of multiplication will not be correct if the digit-sum of the product of the digit-sums is not equal to digit-sum of the product, but the reverse is not true i.e. the answer of multiplication may or may not be correct if the digit-sum of the product of the digit-sums is equal to digit-sum of the product
- Finding the sum of the digits of a number raised to a power
Example: The digits of the number (4)24 are summed up continually till a single digit number is obtained. What is that number?
Answer: 43 = 64. Digit sum of 64 is = 1.
424 = 43 × 43 × 43 … × 43 (8 times)
Digit sums on both sides will be the same.
Þ digit sum of 424 = digit sum of 1 × 1 × 1 × 1… (8 times) = 1
Example: Find the sum of the sum of the sum of the digits of 25!
25! = 1 × 2 × 3 × … × 24 × 25. As one of the multiplicands is 9, the digit sum will be 9.
3. Determining if a number is a perfect square or not
It can be seen from the table that the digit-sum of the numbers which are perfect squares will always be 1, 4, 9, or 7.
Note: A number will NOT be a perfect square if its digit-sum is NOT 1, 4, 7, or 9, but it may or may not be a perfect square if its digit-sum is 1, 4, 7, or 9.
Example: Is the number 323321 a perfect square?
Answer: the digit-sum of the number 323321 is 5. Hence, the number cannot be a perfect square.
Example: A 10-digit number N has among its digits one 1, two 2’s, three 3’s, and four 4’s. Is N be a perfect square?
Answer: We can see that the digit sum of a perfect square is always 1, 4, 7, or 9. As the digit sum of the number is 3, it cannot be a perfect square.
Now can you answer the question that I posed? I bet that you can.
We are done here for today. I hope the reader found this simple concept a useful weapon for his mathematical armoury. It would be an honour if the ghost can help you regarding your queries and doubts. Meet you on the next page after I finish my cigar!