Digit-Sum Method
We’ll look at a quick way to calculate the digit-sum today, as well as how to use it to check the answer involving multiplication, division, addition, subtraction, squares, square roots, cube roots, etc. Instead of quick calculations, this method is utilized for quick response verification. It will enable us to confirm the resolution of a certain query that we have arrived at. Students taking competitive tests should definitely use this strategy.
When examining results including multiplication, division, addition, subtraction, squares, square roots, cube roots, etc., the digit-sum approach, often referred to as the add-up method, can be utilized. The process is extremely straightforward.
Any given number must be reduced to a single digit by repeatedly adding up all of its digits. Let’s use an example to explain.
Find the digit-sum of 2467539.
The number is (2467539). We add the number’s digits one by one.
2 + 4 + 6 + 7 + 5 + 3 + 9 = 36
Now, we take the number 36 and add its digits 3 + 6 to get the answer 9. Thus, we have converted the number 2467539 into its digit-sum 9.
Example 2: Finding the digit-sum of 56768439.
We add all digits of the given number:
5+6+7+6+8+4+3+9 = 48 = 4+8 = 12 =1+2 = 3
Hence, the digit-sum of 56768439 is 3.
(Remember that the sum of the digits will always be one. A single-digit answer can only be obtained by continuing to add the numbers.)
Short-cut method to calculate digit-sum
However, before continuing ahead let’s understand short-cut method to calculate digit sum.
Rule:
While calculating the digit-sum of a number, you can eliminate all the nines and all the digits that add up to nine. When you eliminate all the nines and all the digits that add up to nine you will be able to calculate the digit-sum of any number much faster. The elimination will have no effect on the final result. Let’s understand by an example.
Finding the digit-sum of 637281995.
The digit sum of 6372819923 is: 6+3+7+2+8+1+9+9+2+3 = 50 and again 5+0 is 5.
Now, we will eliminate the numbers that add up to 9 (6 and 3, 7 and 2, 8 and 1 and also eliminate the two 9’s). We are left with the digits 2 and 3 which also add up to 5. Thus, it is established that the shortcut approach can be used to determine the digit-sum. The answer will be the same in either case. Let’s see another example.
Calculating digit sum of 6754231.
Number left after eliminating 9 and digits adding up to 9: 1. I just rearranged the given number to get the sum 9 and eliminate that easily, like 6+3+7+2+5+4+1. So, 1 is the answer.
Note: If the digit-sum of a number is 9, then we can eliminate the 9 straight away and the digit- sum becomes ‘zero.’
We have discussed how to calculate the digit-sum of a number. We’ll now solve a variety of examples involving different arithmetical operations.
Verifying Multiplication By Digit Sum Method
Verify whether 467532 multiplied by 107777 equals 50389196364.
First we will calculate the digit-sum of the multiplicand. Then we will calculate the digit-sum of the multiplier. We will multiply the two digit-sums thus obtained. If the final answer equals to the digit sum of the product then our answer can be concluded to be correct.
The digit sum of 467532 is 9.
The digit sum of 107777 is 2.
When we multiply 9 by 2 we get the answer 18. Again the digit sum of 18 is 9.
Thus, the digit-sum of the completed multiplication procedure is 9.
Now, we will check the digit-sum of the product. The digit sum of 50389196364 is also 9. The digit sum of the question equals to the digit sum of the answer and hence we can assume that the product is correct.
Verify whether 999816 multiplied by 727235 is 727101188760.
The digit sum of 999816 can be instantly found out by eliminating the three 9’s and the combination of 8 plus 1.
The remaining digit is 6 (which becomes our digit-sum).
The digit sum of 727235 can be instantly calculated by eliminating the numbers that add up to nine.
The digit sum of the remaining digits is 8.
When 8 is multiplied by 6 the answer is 48 and the digit sum of 48 is 3.
But, the digit sum of 727101188750 is 2. The digit-sum of the question does not match with the digit-sum of the answer and hence the answer is certainly wrong.
Verifying Division By Digit Sum Method
Verify whether 2308682040 divided by 36524 equals 63210.
We can use the formula that we had learnt in school.
Dividend = Divisor x Quotient + Remainder.
In this case we will be using the same formula but instead of the actual answers we will be using their digit-sums. The digit-sum of Dividend is 6. The digit sum of divisor, quotient and remainder is 2, 3, and 0 respectively. Since 6 = 2 × 3 + 0, we can assume our answer to be correct. In this manner, we can solve other operations too.
Verifying Addition
Verify whether 18273645 plus 9988888 plus 6300852 plus 11111111 is 45674496.
The digit-sum of the numbers is 0, 4, 6 and 8 respectively. The total of these four digit sum is 18 and the digit sum of 18 is 9. The digit sum of 45674496 is also 9 and hence the sum is correct.
