All the techniques that we have discussed till now have emphasized various methods of quick calculation. They have helped us in reducing our time and labour and in some cases provided the final answer without any actual calculation.
In this chapter, we will study the digit-sum method. This method is not used for quick calculation but for quick checking of answers. It will help us verify the answer that we have obtained to a particular question. This technique has wonderful different applications for students giving competitive exams as they are already provided with four alternatives to every answer.
Although the digit-sum method is discussed by Jagadguru Bharati Krishna Maharaj in his study, mathematicians in other parts of the world were aware of this principle even before the thesis of Swamiji was published. Prof. Jackaw Trachtenberg and other mathematicians have dealt with this principle in their research work.
Example: Find the digit-sum of 2467539
(Note: the digit-sum will always be a single digit. You have to keep on adding the numbers until you get a single-digit answer.)
A few more examples are given below:
We have discussed how to calculate the digit-sum of a number. We shall now solve a variety of illustrated examples involving different arithmetical operations.
Example 1 (Multiplication)
(Q) Verify whether 467532 multiplied by 107777 equals 50389196364.
Ans: 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.
Example 2 (Division)
(Q) Verify whether 2308682040 divided by 36524 equals 63210.
Ans: 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 digitsums.
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 sums involving other operations too. However, before continuing ahead I will introduce a further short-cut to this method. The rule says:
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 us take an example:
(Q). Find the digit-sum of 637281995.
Ans: 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. Hence, it is proved that we can use the short-cut method for calculating the digit-sum. The answer will be the same in either case.
A few more examples:
From the above table we can see that the values in column
(b) and the values in column (d) are similar.
Note: If the digit-sum of a number is 9, then we can eliminate the 9 straight away and the digit-sum becomes 0.
Example 3 (Multiplication)
(Q). Verify whether 999816 multiplied by 727235 is 727101188760.
Ans: 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 digitsum of the remaining digits is 8.
When 8 is multiplied by 6 the answer is 48 and the digitsum 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.
Example 4 (Addition)
(Q). Verify whether 18273645 plus 9988888 plus 6300852 plus 11111111 is 45674496.
Ans: 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.
I think four examples will suffice. In a similar manner, we can check the answer obtained by other mathematical operations too. I have given below a list of the mathematical procedures involved in a particular problem and the technique for calculating the digit-sum.
APPLICATIONS
The digit-sum method has immense utility for practitioners of Numerology and other occult sciences. The knowledge that they can eliminate the 9’s and numbers that add up to 9 makes their task simpler.
For students giving competitive and other exams, this technique has a lot of utility. Many times they can check the digit-sum of each of the alternatives with the digit-sum of the question and try to arrive at the correct answer. This will eliminate the need for going through the whole calculation.
However, there is one drawback to this technique. The drawback is that the digit-sum method can tell us only whether an answer is wrong or not. It cannot tell us with complete accuracy whether an answer is correct or not.
This sentence is so important that I would like to repeat it again.
The digit-sum method can only tell us whether an answer is wrong or not. It cannot tell us with complete accuracy whether an answer is correct or not.
Let me illustrate this with an example.
(Q) What is the product of 9993 and 9997.
Method: Assume that you have read the question and calculated the answer as 99900021. The digit-sum of the question is 3 and the digit-sum of the answer is 3 and hence we can assume that the answer is correct.
However, instead of 99900021 had your answer been 99900012 then too the digit-sum would have matched even though the answer is not correct. Or for that matter if your answer would have been 99990021 then too the digit-sum would have matched although this answer is incorrect too. Or in an extreme case, even if your answer would have been 888111021 then still the digit-sum would have matched although it is highly deviated from the correct answer!
Thus, even though the digit-sum of the answer matches with that of the question, you cannot be 100% sure of its accuracy. You can be reasonably sure of its accuracy but cannot swearby it.
However, if the digit-sum of the answer does not match with the digit-sum of the question then you can be 100% sure that the answer is wrong.
In a nutshell,
For practitioners of Numerology and other occult sciences, there is no question of checking answers and hence this technique can per se come to their aid.
EXERCISE
(Note: Before attempting to solve the questions, remember that 9 is synonymous with 0 and can be used interchangeably.)
PART A
Q. (1) Instantly calculate the digit-sum of the following numbers:
(1) 23456789
(2) 27690815
(3) 7543742
(4) 918273645
PART B
Q. (1) Verify whether the following answers are correct or incorrect without actual calculation.
(1) 95123 × 66666 = 6341469918
(2) 838102050 divided by 12345 = 67890
(3) 882 = 7444
(4) 883 = 681472
(5) 475210 + 936111 + 315270 = 726591
(6) 9999999 – 6582170 = 3417829
(7) 6582170 – 9999999 = -3417829
(8) 900 gives quotient 7 and remainder 60120
(9) 0.45632 × 0.65432 = 0.2985793024
PART C
Q. (1) Select the correct answer from among the alternatives without doing actual calculation (calculate the digit-sum of each alternative and match it with the question).
(1) 3569 × 7129 = __
(1) 25443701
(2) 25443421
(3) 25443401
(4) 25445401
(2) 6524 + 3091 + 8254 + 6324 + 7243 + 5111 + 9902 + 3507 = _
(a) 49952
(b) 49852
(c) 59956
(d) 49956
