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.

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