**Digit sum method to Verify the answer**

Here we will learn method which is not any shortcut trick for calculation but **to verify that the answer is correct or wrong**. First, we understand what digit sum method is & after that we will learn how it use. Digit sum method is also known as digit root method.

Digit is any single number from 0 to 9 & **Digit sum is nothing but the sum of all the digits from the given number till we get single digit.**

** **

**e.g. 1) Find the digit sum of 234562**

First, we add all the digits of the given number.

2+3+4+5+6+2 = 22

But digit sum number is always a single digit number, so we add the result’s digits

2 + 2 = 4

**Answer = 4**

** **

**e.g. 2) Find the digit sum of 728154932**

** **7+2+8+1+5+4+ 9+3+2 = 41

Again, add the result,

4 + 1 =5

** Answer = 5 **

** **

**Shortcut Method:**

**While calculating digit sum of any given number ignore the 9 & all the number whose sum is 9. Even after eliminating them there is no effect on the result & we will get our result faster.**

If we check the above example 2, 728154932

7+2+8+1+5+4+ 9+3+2 = 5

Here we ignore the pair of numbers, 7 & 2, 8 & 1, 5 & 4 Whose addition is 9 & also ignore 9 itself.

So, only 2 & 3 are remaining & after adding them we** get 5.**

Means we get same answer.

**Hence, it is proved that in both the way we get same answer.**

**Applications: We use this method to verify that our answer is correct or not by quick way. It is also very useful for the students giving competitive exam, as they need to select correct answer from the given four options. **

By using this method, we can verify the answer of addition, subtraction, multiplication, division, Square- square root & cube- cube root. We will study for all of them one by one.

**a) Addition: The digit sum of all the number used in addition should be equal to the digit sum of answer.**

**Ex. i) 236 + 109 =345**

236 = 2 + 3 + 6 = 11 =1+1=2

109 = 1+0+9 = 10 = 1 + 0 = 1

Now, we add the result,

2 + 1 = **3**

354 = 3 + 5 + 4 =12 = 1+ 2=** 3 **

**As digit sum of both the number & answer is equal i.e. 3**

**Means, our answer is correct.**

**Ex. ii) verify whether 23197534 + 35672309 + 2347920 = 59104635**** **

23197534 = Here we ignore the 9 & the digits whose addition is 9,

Add the remaining numbers,

3+1+3 = 7

Similarly,

35672309 = 5+3+0=8

2347920 = 0

**Total of digit sum of numbers is,**

**7+8+0 = 15 =1+5=6**

**Now, find value of answer,**

**5****9****104635 = 1+0+5 =6**

**As, both digit is 6 means our answer is correct.**

** **

**b) Subtraction: Subtraction of digit sum of smaller number from digit sum of bigger number should be equal to the digit sum of difference.**

** **

**Ex.iii) 2612 -1201 = 1411**

2612 = 11 (Here we take 11, because if we shorten it then digit is 2, & in subtraction we will get negative number.)

1201 = 4

Now, we subtract final value,

11-4 =**7**

Digit sum of answer is,

1411 = 1+4+1+1** = 7 **

**Both the answer is 7.**

**Ex. iv) Verify, 8938275-43525= 8894750**

** **8938275=42

43525 = 19

Subtract final value of the number,

**42- 19 = 23 =2+3=5**

**8894750=41 = 4+1=5**

**Both the answer is, 5 Means sum is correct.**

** **

**C) Multiplication: Multiplication of digit sum of all the number should be equal to digit sum of answer.**

**Ex. v) Verify whether 2435673 x 56442 =1374255466 **

** 2435673 = 3 (Apply shortcut method)**

** 56442 = 6+4+2=12=1+2=3 **

**Multiply them,**

**3×3 = 9 **

**1374255466 =1+6= 7 (apply shortcut method)**

** **

**As both the answer different, means sum is wrong.**

** **

**Ex. vi) 0.45632 x 0.6****5432 = 0.2985793024**

0.45632=0.2

0.65432= 0.2

Multiply them,

0.2 x 0.2 =**0.04**

0.2985793024= **0.04**

**As both the answer is same, means sum is correct. **

** **

**d) Division: To crosscheck the division by digit-sum we must use same formula,**

** Dividend = Divisor x quotient + Remainder **

Instead of number we use digit sum value to check whether answer is correct or not.

**Ex. vii) 2795 ÷ 23, Q = 121, R = 12.**

Dividend, 2795 = 5

Divisor, 23 =5

Q, 121 = 4

R, 12 = 3

Put all the digit sum value in above formula,

Dividend = 5 x 4 + 3 =23 =**5**

**Both the answer is same, means sum is correct.**

**Ex. viii) Verify whether the following answer is correct or not without actual calculation.**

**900 divide by 120 gives Q =7 & R = 60.**

Dividend, 900 = **9**

Divisor, 120 = 3

Q = 7

R = 60 =6

Dividend = 3 x 7 + 6 =27 = **9**

**Both the answer is 9, so without doing actual calculation we can say that the answer is correct.**

** **

**e) Square & Square-Root: Square of number is nothing but multiply a number by same number.**

**Ex. ix) Verify that 25 is square root of 625 or not.**

** Here we apply method of multiplication as**

** We need to check, 25 x 25 = 625 or not.**

First, we find Digit sum of 25×25.

25 = 7

So, 7 x 7 = 49 =13 = **4**

Digit sum of 625 = 13 =** 4**

**As digit of both the side is 4, means 25 is square root of 625.**

** **

**Ex. x) Verify 652 ^{2 }= 425114 or not.**

** **625 = 13 = 4

So, digit sum of 652^{2 }is,

^{ } 4 x 4 =16=**7**

Now value of 425114 is,

4+2+5+1+1+4 = **8**

**As both the digits are not match, so answer is wrong.**

** **

** **

** **

**f) Cube & cube Root: Cube of number is nothing but multiply the same number 3 times, **so here also we apply multiplication method to cross check the answer.

**Ex. x) Verify whether 17 is cube root of 4910 or not.**

** **

Here we need to check, 17x17x17 = 4910 or not.

17 =1+7 = 8

So, 8x8x8=256

Now, we find digit sum of 256

256=13=**4**

similarly, 4910 =14 =**5**

As, digit of both the side is not same, so even without solving sum we can say that,

**17 is not the cube root of 4910. **

** **

**Note: One thing I would like to mention here that, by using digit sum method we can say that answer is wrong or not, ****but we are not 100% confirm about answer is correct or not. **

**e.g. 2345 x 645 = 1512525**

Here digit sum of question is 3 & digit sum of answer is also 3 means answer is correct.

But even if we get answer 1512552 or 1512255 then also the digit sum of answer will be 3 which is matched with questions digit sum, but the answer is incorrect here.

**Find this links to study more:**

Base method for easy multiplication

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