What is Digit Sum?
For a given number digit sum is the sum of all its digits. If the sum is greater than 9 then its digits are again added to get the new sum until its sum lies between 1 and 9 i.e. a single digit number.
Examples:
Number
|
Digit Sum
|
Casting out 9’s:
When finding the Digit Sum of a number, 9’s can be “cast out”. By Casting Out 9’s, we can find the Digit Sum more quickly and mentally.
Example: 94993
When finding the Digit Sum of a number, 9’s can be “cast out”. By Casting Out 9’s, we can find the Digit Sum more quickly and mentally.
Example: 94993
Digit Sum: 4+3 = 7
If the number has a group of numbers that sum to 9 can be “cast out”
Example: 549673
Digit Sum: 7
Digit Sum: 7
“cast out” the 5+4, 9 and 6+3, leaving just 7
QUICK CHECKING OF ANSWER USING DIGIT SUM
ADDITION
Rule: Digit sum of the result of the addition must equal the digit sum of result obtained by addition of digit sum of individual numbers.
Problem
|
Digit Sum
| |
143
|
1+4+3 =
|
8
|
+ 302
|
3+0+2 =
|
5
|
+ 467
|
4+6+7=17, 1+7 =
|
8
|
912
|
21
| |
9+1+2=12, 1+2=3
|
2+1=3
|
SUBTRACTION
Rule: Subtract the Digit Sums of the numbers in the problem; the resulting Digit Sum must equal the Digit Sum of the of the answer.
Problem Digit Sum Check
776 2 - 152 - 8* 624 3
Digit Sum of the answer = 3
| 776 reduces to 2 152 reduces to 8 (2 + 9*)- 8 = 3 Answer 624 reduces to 3 In this example it's clear that 8 cannot be subtracted from 2. *Solution: Add 9 to the smaller number above and subtract normally. |
In the above example, an alternative would be to add the numbers starting at the bottom. Add 8 + 3 to equal 11, which would reduce to 2.
Problem Digit Sum Check
2,489 5 - 382 - 4 - 932 - 5* 1,175 5
Digit Sum of the answer = 5
| 2,489 reduces to 5 382 reduces to 4 932 reduces to 5 (5 - 4)+ 9* - 5 = 5 Answer 1,175 reduces to 5 In this example, it's possible to subtract 4 from 5 but we cannot subtract 5 from 1. *Solution: Once again, add 9 to the smaller number above and subtract normally. |
Once again an alternative would be to add the numbers starting at the bottom. Add 5 + 5 + 4 = 14, which reduces to 5.
MULTIPLICATION
Terminology: Multiplicand, Multiplier and Product
In the example 3 x 2 = 6, 3 is the multiplicand, 2 is the multiplier and 6 is the product.
In the example 3 x 2 = 6, 3 is the multiplicand, 2 is the multiplier and 6 is the product.
Rule: Multiply the Digit Sum of the multiplier x The Digit Sum of the multiplicand; the resulting Digit Sum must equal the Digit Sum of the of the product.
Problem Digit Sum Check
274 4 x 51 x 6 13,974 6
Digit Sum of the product = 6
|
2+7+4 = 13, 1+3=4
5+1=6 4 x 6 = 24, reduces to 6, also Product 13,974 reduces to 6 |
Problem Digit Sum Check
3,875 5
x 834 x 6 3,231,750 3
Digit Sum of the product = 3
|
Multiplicand 3,875 reduces to 5
Multiplier 834 reduces to 6 5 x 6 = 30, reduces to 3, Product 3,231,750 reduces to 3 |
DIVISION
Terminology: Dividend, Divisor and Quotient
In the problem 12 ÷ 3 = 4, 12 is the dividend, 3 is the divisor and 4 is the quotient.
In the problem 12 ÷ 3 = 4, 12 is the dividend, 3 is the divisor and 4 is the quotient.
Rule: Multiply the Digit Sum of the quotient x the Digit Sum of the divisor; the resulting Digit Sum must equal the Digit Sum of the dividend.
Digit Sum of 147 is 3 ( 1+4+7=12, 1+2=3)
Calculating Digit Sum of 49x3 :
Digit sum of 49 is 4
Digit sum of 3 is 3
Digit sum of 49x4 = 4x3= 12, 1+2=3
Thus we verified the above rule.
Applications of Digit-Sum in checking if a number is a perfect square or not:
No
|
Square
|
Last Digit
|
Digit Sum
|
No
|
Square
|
Last Digit
|
Digit Sum
|
1
|
1
|
1
|
1
|
10
|
100
|
0
|
1
|
2
|
4
|
4
|
4
|
11
|
121
|
1
|
4
|
3
|
9
|
9
|
9
|
12
|
144
|
4
|
9
|
4
|
16
|
6
|
7
|
13
|
169
|
9
|
7
|
5
|
25
|
5
|
7
|
14
|
196
|
6
|
7
|
6
|
36
|
6
|
9
|
15
|
225
|
5
|
9
|
7
|
49
|
9
|
4
|
16
|
256
|
6
|
4
|
8
|
64
|
4
|
1
|
17
|
289
|
9
|
1
|
9
|
81
|
1
|
9
|
18
|
324
|
4
|
9
|
Important Points:
- Last digit of perfect square number can have 1, 4, 9, 6, 5 or 0.
- Digit sum of the numbers which are perfect squares can have 1, 4, 9 or 7.
Example: Is the number 323321 a perfect square?
Answer: the digit-sum of the number 323321 is 5. Hence, the number cannot be a perfect square.
Example: Is the number 71289 a perfect square?
Answer: the digit-sum of the number 71289 is 9. Hence, the number may be a perfect square.
Let's try to solve some questions which have appeared in competitive exams like IBPS, SSC etc.
Q. What will appear in place of question mark (?) in the following questions
1). 4003 x 77 - 21015 = ? x 116
(a) 2477 (b) 2478 (c) 2467 (d) 2476
Solution: In first step find the digit sum of all numbers in the expression
Digit Sum of 4003 = 7
Digit Sum of 77 = 5
Digit Sum of 21015 = 9 or 0
Digit Sum of 116 = 8
Now solve the above expression using digit sum method
Digit Sum of ( 4003 x 77 - 21015) = (digit sum of 4003) x (digit sum of 77) - (digit sum of 21015)
= 7 x 5 - 9
= 35 - 9
= 8 - 9
= - 1 or 8 ( if -ve sign, then subtract it from 9)
We calculated digit sum of left side of expression which is equal to 8
Then right side of its expression should also have digit sum = 8
Hence the number at place of question mark must have digit sum = 1 in order to make right side of expression's digit sum = 8
In next step we have to check digit sum of the options to find the correct answer
(a) Digit Sum of 2477 = 2
(b) Digit Sum of 2478 = 3
(c) Digit Sum of 2467 = 1
(d) Digit Sum of 2476 = 1
It is clear that options (a) and (b) are incorrect, and options (c) and (d) both have digit sum =1 ( what we were expecting)
To choose correct option we use a very simple method
Just check the last digit of both sides of expression
Last digit of left side expression 4003 x 77 - 21015 = 3 x 7 - 5 = 6
Last digit of right side of expression will be 6 if multiplied by 2476
Hence correct option is (d) 2476
Solution: First solve (5√7 + √7) × (4√7 + 8√7) = 6√7 × 12√7 = 72 × 7
Digit Sum of 72 × 7 = 9
Digit Sum of 19² = 19 × 19 = 1 × 1 = 1
Digit Sum of left side of expression = 9 − 1 = 8
Clearly option (1) and (3) have digit sum = 8
Again to eliminate one option use very simple method, just check the result is even or odd 72 x 7 is even, 19² is odd, and their difference will be odd
Hence correct option is (1) 143
Let's try to solve some questions which have appeared in competitive exams like IBPS, SSC etc.
Q. What will appear in place of question mark (?) in the following questions
1). 4003 x 77 - 21015 = ? x 116
(a) 2477 (b) 2478 (c) 2467 (d) 2476
Solution: In first step find the digit sum of all numbers in the expression
Digit Sum of 4003 = 7
Digit Sum of 77 = 5
Digit Sum of 21015 = 9 or 0
Digit Sum of 116 = 8
Now solve the above expression using digit sum method
Digit Sum of ( 4003 x 77 - 21015) = (digit sum of 4003) x (digit sum of 77) - (digit sum of 21015)
= 7 x 5 - 9
= 35 - 9
= 8 - 9
= - 1 or 8 ( if -ve sign, then subtract it from 9)
We calculated digit sum of left side of expression which is equal to 8
Then right side of its expression should also have digit sum = 8
Hence the number at place of question mark must have digit sum = 1 in order to make right side of expression's digit sum = 8
In next step we have to check digit sum of the options to find the correct answer
(a) Digit Sum of 2477 = 2
(b) Digit Sum of 2478 = 3
(c) Digit Sum of 2467 = 1
(d) Digit Sum of 2476 = 1
It is clear that options (a) and (b) are incorrect, and options (c) and (d) both have digit sum =1 ( what we were expecting)
To choose correct option we use a very simple method
Just check the last digit of both sides of expression
Last digit of left side expression 4003 x 77 - 21015 = 3 x 7 - 5 = 6
Last digit of right side of expression will be 6 if multiplied by 2476
Hence correct option is (d) 2476
Solution: First solve (5√7 + √7) × (4√7 + 8√7) = 6√7 × 12√7 = 72 × 7
Digit Sum of 72 × 7 = 9
Digit Sum of 19² = 19 × 19 = 1 × 1 = 1
Digit Sum of left side of expression = 9 − 1 = 8
Clearly option (1) and (3) have digit sum = 8
Again to eliminate one option use very simple method, just check the result is even or odd 72 x 7 is even, 19² is odd, and their difference will be odd
Hence correct option is (1) 143
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