Divisibility of Numbers

Tests of divisibility:
There are certain tests of divisibility that can help us to decide whether a given number is divisible by another number without actual division.

Divisibility of numbers by 10:
A number that has 0 in its ones place is divisible by 10.

Divisibility of numbers by 5:
A number that has either 0 or 5 in its ones place is divisible by 5.

Divisibility of numbers by 2:
A number that has 0, 2, 4, 6 or 8 in its ones place is divisible by 2.

Divisibility of numbers by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility of numbers by 6:
A number is divisible by 6 if that number is divisible by both 2 and 3.

Divisibility of numbers by 4:
A number is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.

Divisibility of numbers by 8:
A number is divisible by 8 if the number formed by its last three digits (i.e. ones, tens and hundreds) is divisible by 8.

Divisibility of numbers by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility of numbers by 11:
If the difference between the sum of the digits at the odd and even places in a given number is either 0 or a multiple of 11, then the given number is divisible by 11.

Co–prime numbers:
If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.

General rules of divisibility for all numbers
If a number is divisible by another number, then it is also divisible by all the factors of the other number. If two numbers are divisible by another number, then their sum and difference are also divisible by the other number. If a number is divisible by two co-prime numbers, then it is also divisible by the product of the two co-prime numbers.

Summary

Tests of divisibility:
There are certain tests of divisibility that can help us to decide whether a given number is divisible by another number without actual division.

Divisibility of numbers by 10:
A number that has 0 in its ones place is divisible by 10.

Divisibility of numbers by 5:
A number that has either 0 or 5 in its ones place is divisible by 5.

Divisibility of numbers by 2:
A number that has 0, 2, 4, 6 or 8 in its ones place is divisible by 2.

Divisibility of numbers by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility of numbers by 6:
A number is divisible by 6 if that number is divisible by both 2 and 3.

Divisibility of numbers by 4:
A number is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.

Divisibility of numbers by 8:
A number is divisible by 8 if the number formed by its last three digits (i.e. ones, tens and hundreds) is divisible by 8.

Divisibility of numbers by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility of numbers by 11:
If the difference between the sum of the digits at the odd and even places in a given number is either 0 or a multiple of 11, then the given number is divisible by 11.

Co–prime numbers:
If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.

General rules of divisibility for all numbers
If a number is divisible by another number, then it is also divisible by all the factors of the other number. If two numbers are divisible by another number, then their sum and difference are also divisible by the other number. If a number is divisible by two co-prime numbers, then it is also divisible by the product of the two co-prime numbers.

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