# What Have We Discussed?

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We have discussed multiples, divisors, factors and have seen how to identify factors and multiples.

**We have discussed and discovered the following :**

(a) A factor of a number is an exact

(b) **Every number** is a factor of itself.

(c) **Every factor** of a number is less than or

(d) **Every number** is a

(e) **Every multiple** of a given number is

(f) **Every number** is a

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- We have learnt that –

(a) The number other than 1, with only factors namely 1 and the number itself, is a **neither prime nor composite.**

(b) The number 2 is the smallest

(c) Two numbers with only 1 as a common factor are called

(d) A number divisible by **two co-prime numbers** is divisible by their

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- We have discussed how we can find just by looking at a number, whether it is divisible by small numbers 2,3,4,5,6,7,8,9 and 11. We have explored the relationship between digits of the numbers and their
**divisibility by different numbers.**

**Divisibility by 2 :**- If the number ends in an even digit (0, 2, 4, 6, 8), it is divisible by 2.
- Can be seen by just the

**Divisibility by 3 :**- Add up all the digits of the number. If the sum is divisible by 3, then the original number is also divisible by 3.
- checked by finding the
of all digits

**Divisibility by 4 :**- Check if the last two digits of the number form a number divisible by 4. If they do, then the original number is divisible by 4.
- is checked by the last
digits respectively.

**Divisibility by 5 :**- If the number ends in 0 or 5, it is divisible by 5.
- can be seen by just the
.

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**Divisibility by 6 :**- A number is divisible by 6 if it is divisible by both 2 and 3.

**Divisibility by 7 :**- The divisibility rule for 7 is more complex. You can use the "subtract and divide by 7" method.
- Take the last digit, double it, and subtract the result from the remaining part of the number.
- If the result is divisible by 7 or is 0
- then the original number is divisible by 7.

**Divisibility by 8 :**- Check if the last three digits of the number form a number divisible by 8. If they do, then the original number is divisible by 8.
- is checked by the last
digits respectively.

**Divisibility by 9 :**- Add up all the digits of the number. If the sum is divisible by 9, then the original number is also divisible by 9.
- checked by finding the
of all digits.

**Divisibility by 10 :**- If the number ends in 0, it is divisible by 10.

**Divisibility by 11 :**- Subtract the alternating sum of the digits (starting from the left) from the alternating sum of the digits (starting from the right).
- If the result is divisible by 11, then the original number is divisible by 11.

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**We have learnt that**

(a) The Highest Common Factor (HCF) of two or more given numbers is the

(b) The Lowest Common Multiple (LCM) of two or more given numbers is the