# What does the sum refer to in math

## Divisor of a number

If you can divide a number a by a number b with no remainder, then a is divisible by b.
One then also says: b is the divisor of a

 Example: 6 is a divisor of 18, because 18: 6 = 3 remainder 0 6 is not a divisor of 17, because 17: 6 = 2 remainder 5

To find out whether a number b is a divisor of a second number a, there are some rules:

### 1. Sum (difference) rule

When a number divides two other numbers, it also divides the sum or difference of those numbers.

 Example: 6 is a factor of 18 and 6 is a factor of 720. So 6 is also a divisor of 720 + 18 = 738. 6 is a factor of 720, but 6 is not a factor of 17. So 6 is not a divisor of 720 + 17 = 737 either. 7 is a factor of 700 and 7 is a factor of 21. So 7 divisors of 700-21 = 679. 7 is a divisor of 1400 and 7 is not a divisor of 15. So 7 is not a divisor of 1400-15 = 1385.

In order to use this rule to examine whether a number a is a divisor of a number b, the number b is broken down into a sum or difference in such a way that it is easy to determine whether a is a divisor of both summands or of the minuend and subtrahend is. If this is the case, then a is a factor of b, otherwise not.

When breaking it down into a sum or difference, it is essential to ensure that at least one of the two summands or the minuend or the subtrahend is definitely divisible by a.

 Example: Is 8 a divisor of 2464? Divide 2464 into a suitable sum, e.g .: 2464=2400+64 Since 8 is a divisor of 2400 and 64, 8 is also a factor of 2464. Is 15 a divisor of 1475? 1575 is broken down into a suitable difference, e.g .: 1475=1500-25 Since 15 is a factor of 1500 but not a factor of 25, 15 is not a factor of 1475

Caution: The reverse conclusion (a number a is not divisible by a number b if you can divide b in such a way that none of the summands is a divisor of a) applies Not!

Example: It's 14 = 9 + 5. While neither 9 nor 5 is divisible by 7, 14 is divisible by 7.

### 2nd divisor rule

If a number a is a divisor of a number b, then every divisor of a is also a divisor of b.

 Example: Since 12 is a divisor of 144, all are divisors of 12, i.e. 1; 2; 3; 4; 6 divisors of 144.

### 3. Terminal rules

 A number is divisible by 10 if its last digit is 0. A number is divisible by 5 if its last digit is a 5 or a 0. A number is divisible by 2 if its last digit is an even digit A number is divisible by 4 if its last two digits represent a number divisible by 4 A number is divisible by 8 if its last three digits represent a number that is divisible by 8.
 Example: Which of the numbers 2, 4, 5, 8, 10 is 13740 divisible by? 13740 is divisible by 2 because the last digit is an even digit. 13740 is divisible by 4 because the last two digits represent a number that is divisible by 4, namely 40. 13740 is divisible by 5 because the last digit is 0. 13740 is not divisible by 8 because the last three digits are not a number that can be divided by 8. 13740 is divisible by 10 because the last digit is 0.

### 4. checksum rules

 A number is divisible by 3 if its cross sum (i.e. the sum of all digits in the number) is divisible by 3. A number is divisible by 9 if its cross sum (i.e. the sum of all digits in the number) is divisible by 9.
 Example: Which of the numbers 3 and 9 is 13740 divisible by? 13740 is divisible by 3 because its checksum (1 + 3 + 7 + 4 + 0 = 15) is divisible by 3.13740 is not divisible by 9 because its checksum (1 + 3 + 7 + 4 + 0 = 15) is not divisible by 9.