What Have We Discussed?

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

2. We have discussed and discovered the following :

(a) A factor of a number is an exact divisormultiple of that number.

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

(c) Every factor of a number is less than or equalnot equal to the given number.

(d) Every number is a multipledivide of each of its factors.

(e) Every multiple of a given number is greaterlesser than or equal to that number.

(f) Every number is a factor and multiple multiplefactor of itself.

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3.We have learnt that –

(a) The number other than 1, with only factors namely 1 and the number itself, is a prime numberfactor. Numbers that have more than two factors are called composite numbersreal numbers. Number 13 is neither prime nor composite.

(b) The number 2 is the smallest primefactor and is even. Every prime number other than 2 is oddeven.

(c) Two numbers with only 1 as a common factor are called co-prime numbers.

(d) A number divisible by two co-prime numbers is divisible by their product sumdifference also.

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4.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 last digitfirst digit

• 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 summultiple 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 last digitfirst digit.

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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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5. We have learnt that

(a) The Highest Common Factor (HCF) of two or more given numbers is the highest lowest of their common factors.

(b) The Lowest Common Multiple (LCM) of two or more given numbers is the lowest highest of their common multiples.