What is the Digit Sum?
Given a number N1, all the digits of N1 are added to obtain a number N2. 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
What are the uses of the concept of the Digit Sum?
We introduce you to a very important concept of the Digit-Sum Rule of Multiplication here. This rule says that: 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.
Solution: We will check this using the rule of Digit Sum rule of Multiplication.
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
Hence, Digit sum of the product (4515) = Digit-sum of the product of the digit sums (24) = 6
A very important use of this concept in Problem solving includes:
1. Rapid checking of calculations while multiplying numbers
If you want to check the multiplication of different numbers is correct or not, you can use this method to check the same. Suppose you are taking ou the product of AxBxC = D. To check you need to take out the digit sum of the product number and the individual numbers. If both the value matches then the product is taken out will be correct, else not.
- 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, 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.
2. Finding the sum of the digits of a number raised to a power
Example: The digits of the number (4)30 are summed up continually till a single-digit number is obtained. What is that number?
Answer: 43 = 64. Digit sum of 64 is = 1.
430 = 43 × 43 × 43 ... × 43 (10 times)
Digit sums on both sides will be the same.
--> digit sum of 430 = digit sum of 1 × 1 × 1 × 1... (10 times) = 1
3. Determining if a number is a perfect square or not
To understand this concept let us see the square of the first 18 numbers, along with their Digit sum
From the above table, we see that the digit sum rule of all the squares come out to be 1, 4, 7 or 9. Hence the rule is very clear from here that the digit sum of all the Perfect Squares will be 1, 4, 7 or 9. Hence, to check whether a given number is square or not you can check the digit sum of the number and if it comes out to be any one of the numbers from 1, 4, 7 or 9 then the number is a perfect square else not.
- 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.
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