Rule -1 We have already discussed that the digit sum can be any number from 1 to 9 i.e. digit sum can be 1, 2, 3, 4, 5, 6, 7, 8, 9. If we add 9 to any of these numbers the digit sum will not change means if we add 1 + 9 = 10, the digit sum will be 1 + 0 = 1, similarly 5 + 9 = 14, and the digit sum will be 1 + 4 = 5. Hence, we can say the digit sum of 4578 will be the same as the digit sum of 45789 or 94578 etc, thus while calculating the digit sum of any number we keep on eliminating 9 from the number, this process helps in speedy calculation of the digit sum. Like for calculating the digit sum of 45869786, we calculate the digit sum of 8 + 6 + 7 + 8 + 6 (45869786) 4 and 5 are also eliminated as the sum of 4 and 5 is 9. Similarly you can eliminate any pair from the number which sums to 9 like pair (8 & 1, 7 & 2, 6 & 3, 5 & 4)
Rule-2 We can calculate the digit sum of any relative value like percentage or fraction values like decimal by just removing percentage sign or decimal from the number and calculating the digit sum of the residual number. like the digit sum of 78% is the same as 78 or the digit sum of 455876.5 is the same as 4558765.
Rule -3 The Digit sum of any multiple of 9 is always 9 means the digit sum of 9 x expression is 9 like the digit sum of 9 x 5578241 is 9. Here you should note that the expression can be any expression provided 9 in the numerator should not cancel out from the denominator like the digit sum of \({9 \times 5578241} \over 7\) is 9 but if in the denominator we have 3 or 9 in place of 7 the rule fails and we have to calculate the digit sum by eliminating 3 or 9 from the denominator, which we have discussed in other. This rule helps in the quick calculation of the digit sum of an expression.
Rule -4: Digit Sum of Negative Number
The digit sum of a -ve number is calculated similarly as it is calculated for a +ve number. However, after calculating the digit sum of the number a -ve sign is placed before the digit sum and then the sign of the digit sum is changed by subtracting the digit sum from 9. i.e. the digit sum of -2714 is calculated as - (digit sum of 2714) which is -5 and then the -ve sign of the digit sum is changed as 9 - 5 = 4. Hence, the digit sum of -2714 is 4 or -5.
How to change the sign of digit sum. If the digit sum of any expression turns out to be -ve we can easily change it to -ve by subtracting the -ve digit sum from 9. i.e. if the digit sum of any expression is -7 then to make it +ve we should simply subtract -7 from 9 to get the digit sum i.e. 9 -7 = 2 (here, 2 will be the digit sum)
GENERAL RULES
Addition
The digit sum of the expression with the addition of two or more numbers is equal to the digit sum of the output of the expression i.e. the digit sum of 564658 + 125468 + 45328 + 14224 is 5 ⇒ (7 + 8 + 4 + 4 = 23 ⇒ 2 + 3 = 5) and the output of expression is 749678, the digit sum of which is also 5. Hence, if you have to calculate the sum of 564658 + 125468 + 45328 + 14224 and you are provided with options (749678, 749978, 749878, 749778), then instead of doing actual addition, you can simply check the digit sum of the expression 564658 + 125468 + 45328 + 14224 and the options and accordingly select the correct answer. This process will certainly help you to save considerable time.
Note: A similar rule applies in the case of subtraction and multiplication. examples:
Subtraction
a) 8978 - 5455 - 809 = 2714
the digit sum of 8978 - 5455 - 809 is -4 or 5 ⇒ (5 - 1 - 8). Since -4 is a negative number, we should change -4 to a positive digit sum by subtracting the digit sum from 9 i.e. 9 - 4 = 5. Hence the digit sum of expression 8978 - 5455 - 809 is 5 and the digit sum of the output is 2714 is also 5
b) 5455 + 809 - 8978 = -2714
In this case, the digit sum of 5455 + 809 - 8978 is 4 and the digit sum of -2714 is -5, which is -ve and hence we should remove the -ve sign by subtracting 5 from 9 i.e. 9 - 5 = 4 hence the digit sum of -2714 is also 4.
From, the above we should understand in case of subtraction the digit sum may come to -ve and we should not forget to change it +ve while evaluating the correct answer
Multiplication
If there is a need to calculate the value expression 14 x 25 x 40 x 31, then we can calculate the digit sum by replacing the numbers with its digit sum and then proceed as below
14 x 25 x 40 x 31 → 5 x 7 x 4 x 4 → 35 x 4 x 4 → 8 x 4 x 4 → 32 x 4 → 5 x 4 → 20 → 2 (digit sum) [you can multiply digit sum in any order]
now if we check the digit sum of 4340000 (14 x 25 x 40 x 31) it will also come out to be 2
Division
The calculation of the digit sum when division is involved in an expression may appear tricky, as we have to separately calculate the digit sum of the numerator and denominator.
- The initial step should be the calculation of the digital sum of the denominator which can be from 1 to 9.
- Now if the digital sum of the denominator is 3, 6 or 9 then, instead of calculating the digital sum of the numerator, we should try to eliminate factor 3 or 9 from the numerator and denominator.
Note the elimination should be from the actual expression like if we have to calculate the digital sum of \(5202 \over 153\), the first step should be to calculate the digital sum of the denominator i.e. 153 which comes to be 9. Now since 9 is in the denominator we should try to eliminate this 9 from the denominator and numerator by calculating the factor thus reducing \({5202 \over 153 } = {578 \over17}\)
-Now again calculate the digit sum of the denominator i.e. 17, if the digit sum is again 3, 6, or 9 try to further eliminate them. If the factor 3 or 9 cannot be eliminated we cannot find the digital sum of expression.
If the digit sum of the denominator is other than 6, 6 or 9 we can proceed further as below:
♦\(digit \space sum \space of \space Nr \over 1 \)→ the digit sum of expression is the digit sum of Nr
like the digit sum of \(323 \over 19 \) is 8 as the digit sum of Dr is 1
Now in all other cases where the digit sum of the denominator is 2, 4, 5, 7, 8 we will try to make the digit sum of the denominator as 1
♦\(digit \space sum \space of \space Nr \over 2\) → multiply both Nr and Dr by 5 → \(digit \space sum \space of \space Nr \space \times 5 \over 2 \space \times 5\) → the digit sum of expression
like the digit sum of \(341 \over 11 \)is 8 x 5 → 4 → as the digit sum of Dr is 2
♦\(digit \space sum \space of \space Nr \over 4\) → multiply both Nr and Dr by 7 → \(digit \space sum \space of \space Nr \space \times 7 \over 4 \space \times 7\) → the digit sum of expression
like the digit sum of \(299 \over 13\)is 2 x 7 → 5 → as the digit sum of Dr is 4
♦\(digit \space sum \space of \space Nr \over 5\) → multiply both Nr and Dr by 2 → \(digit \space sum \space of \space Nr \space \times 2 \over 5 \space \times 2\) → the digit sum of expression
like the digit sum of \(322 \over 14\)is 7 x 2 → 5 → as the digit sum of Dr is 5
♦\(digit \space sum \space of \space Nr \over 7\) → multiply both Nr and Dr by 4 → \(digit \space sum \space of \space Nr \space \times 4 \over 7 \space \times 4\) → the digit sum of expression
like the digit sum of \(368 \over 16\)is 8 x 4 → 5 → as the digit sum of Dr is 7
♦\(digit \space sum \space of \space Nr \over 8\) → multiply both Nr and Dr by 8 → \(digit \space sum \space of \space Nr \space \times 8 \over 8 \space \times 8\) → the digit sum of expression
like the digit sum of \(391 \over 17\)is 4 x 8 → 5 → as the digit sum of Dr is 8
An example with detailed explanation
like in \({5202 \over 153 } \) we have first eliminated 9 from the denominator and reduce it to \({578 \over17}\), now in the denominator, we have 17 and its digital sum is 8 thus we have expression like \(digit \space sum \space of \space Nr \over 8\). Now the digit sum of Nr is 2 (5 + 7 + 8) and thus we can reduce equation as \(2 \over 8\) and multiply with 8 to make the digit sum of Dr as 1 like \({2 \times 8 \over 8 \times 8 } = {16 \over 64}\) by this way we can reduce the digit sum of ((Dr as 1 (6 + 4). Thus the digit sum of expression \({5202 \over 153 } \) is 7 (1 + 6). Now we can check with options, in actual the value of \({5202 \over 153 } = 34\) and the digit sum of 34 is 7 (3 + 4).
Note: In case of division never cancel the digit sum from the numerator and denominator directly like in above example we should never cancel \(2 \over 8\) and reduce it to \(1 \over 4\). We should always keep in mind that though we can perform addition/ substraction/ multiplication of digit sum as we do with actual no, we should avoid dividing digit sum as it may result in an error in some cases.
Note: If factor 3, 6 or 9 cannot be eliminated from the denominator then the digit sum rule will not apply means we cannot calculate the digit sum of the expression. For example we cannot calculate the digit sum of \(1 \over 3\), \(160 \over 9 \), \(37 \over 6\) etc.
Perfect Square/Square Root
We now understand that any number in the number system has a digit sum ranging from 1 to 9 and we also know that the rule of normal multiplication applies in the digit sum means 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16 → 7 (as 7 is the digit sum), 5 x 5 = 25 → 7, 6 x 6 = 36 → 9, 7 x 7 = 49 → 4, 8 x 8 = 64 → 1, 9 x 9 = 81 → 9. If you put your brain to work you can easily interpret that this will happen when we calculate the square of number. Thus we can say a number cannot be a perfect square if its digit sum is other than 1, 4, 7 or 9 this rule should not be confused with a rule that a number cannot be a perfect square if its unit digit is other than 1, 4, 5, 6 or 9.
Perfect Cube/Cube Root
Similarly, we can extend the multiplication of the digit sum rule for calculating cube or higher power even. For calculating cube you can check by multiplying the unit sums like 1 x 1 x 1 = 1, 2 x 2 x 2 = 8, 3 x 3 x 3 = 9, 4 x 4 x 4 = 1, 5 x 5 x 5 = 8, 6 x 6 x 6 = 9, 7 x 7 x 7 = 1, 8 x 8 x 8 = 8, 9 x 9 x 9 = 9. Hence, we can say a number cannot be a perfect cube if its digit sum is other than 1, 8 or 9.
We have covered all the primary rules related to the calculation of digit sum, now we are giving you some examples where you can directly use these rules and feel the magic of DSM (digit sum method).