Playing with Numbers

**Category : **6th Class

**Playing with Numbers**

- A factor of a number is an exact divisor of that number.

- A number is said to be a multiple of any of its factors.

**e.g..** We know that 35 = 1\[\times \]35 and 35 = 5 \[\times \] 7.

This shows that each of the numbers 1, 5, 7 and 35 divides 35 exactly.

Therefore 1, 5, 7 and 35 are all factors of 35 and 35 is a multiple of each one of the numbers

1, 5, 7 and 35.

- All multiples of 2 are called even numbers.

**e.g.,** 2, 4, 6,8,10, etc.

- Numbers which are not multiples of 2 are called odd numbers.

**e.g.,** 1,3,5,7,9,11, etc.

- Each of the numbers which has exactly two distinct factors, namely 1 and itself is called a

prime number.

**e.g.,** 2,3,5,7,11,13,17,19,23,29 etc.

- Numbers having more than two factors are known as composite numbers.

**e.g.,** 4, 6, 8,9,10 etc.

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**Note: (i) 1 is neither prime composite. (since 1=1, the two factors are not distinct.)**

**(ii) 2 is the lowest prime numbers.**

**(iii) 2 is the only even prime number. (All other even number are composite numbers.)**

- Two consecutive prime numbers differing by 2 are known as twin-primes.

**e.g.,** (i) 3, 5 (ii) 5, 7 (iii) 11, 13 etc.

A set of three consecutive prime numbers, differing by 2, is called a prime triplet.

An example of prime triplet is (3, 5, 7).

- If the sum of all the factors of a number is twice the number then the number is called a perfect number,

**e.g.,** 6 is a perfect number, since the factors of 6 are 1,2,3,6 and (1 + 2 + 3 + 6) = (2 x 6).

- Two numbers are said to be co-prime if they do not have a common factor other than 1.

**e.g.,** (i) 2, 3 (ii) 3, 4 (iii) 8, 15

**Note: (i) Two prime numbers are always co – prime.**

**(ii) Two co – prime need not be prime numbers**.

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**e.g.,** 6, 7 are co-primes, while 6 is not a prime number

- Every even number greater than 4 can be expressed as the sum of two odd prime numbers.

**e.g.,** (i) 6 = 3 + 3 (ii) 8 = 3+5

**Tests of divisibility of numbers**

**Test of divisibility by 2:**

- A number is divisible by 2, if its units digit is 0,2,4,6 or 8.

**e.g.,** 42,84,120,1456,568 etc. are divisible by 2.

**Test of divisibility by 3:**

A number is divisible by 3, if the sum of its digits is divisible by 3.

**e.g.,** Consider the number 64752.

Sum of its digits = (6 + 4 + 7 + 5 + 2) = 24, which is divisible by 3.

\[\therefore \]64752 is divisible by 3.

**Test of divisibility by 4:**

A number is divisible by 4, if the number formed by its digits in tens and units places is divisible by 4.

**e.g.,** Consider the number 49812.

The number formed by tens and units digit is 12, which is divisible by 4.

\[\therefore \]49812 is divisible by 4.

**Test of divisibility by 5:**

A number is divisible by 5, if its unit’s digit is 0 or 5.

**e.g.,** 15, 35, 80,90,1435 etc., are divisible by 5.

**Test of divisibility by 6:**

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- A number is divisible by 6, if it is divisible by both 2 and 3.

**e.g.,** Consider the number 26574.

Its units digit is 4. So it is divisible by 2.

Sum of its digits =2+6+5+7+4= 24, which is divisible by 3.

Therefore, 26574 is divisible by both 2 and 3.

And hence it is divisible by 6.

- Test of divisibility by 8:

A number is divisible by 8, if the number formed by its digits in hundreds, tens and unit's places is divisible by 8.

**e.g.,** Consider the number 47192.

- The number formed by hundreds, tens and units digits is 192, which is divisible by 8.

So, 47192 is divisible by 8.

**Test of divisibility by 9:**

A number is divisible by 9, if the sum of its digits is divisible by 9.

**e.g..** Consider the number 72306.

Sum of its digits =7+2+3+0+6 = 18, which is divisible by 9. Therefore, 72306 is divisible by 9.

**Test of divisibility by 10;**

A number is divisible by 10, if its units digit is zero.

**e.g.,** 10,20,130,580,700,7050, etc., are divisible by 10.

**Test of divisibility by 11:**

A number is divisible by 11, if the difference of the sum of its digits in odd places and sum of its digits in even places (starting from units place) is either 0 or a multiple of 11.

**e.g.,** Consider the number 530728.

Sum of the digits in odd places = 8 + 7 + 3 = 18

Sum of the digits in even places = 2 + 0 + 5 = 7

Difference of these sums = 18 – 7 = 11 which is divisible by 11.

Therefore, 530728 is divisible by 11.

- Every composite number can be factorised into primes in only one way except for the order of primes. This property is known as unique factorization property.

**Least Common Multiple (L.C.M.)**

The Least Common Multiple (L.C.M.) of given numbers can be found by the following methods.

**(i) By Listing Multiples:** List the multiples of the given numbers and then find the least

of the common multiples.

(ii) Multiples of 36 are 36, ** 72**, 108,

Multiples of 72 are ** 72**,

Observe that the common multiples are highlighted.

\[\therefore \]The least of the common multiples is 72. Hence L.C.M. of 36 and 72 is 72.

**By Prime Factorisation Method (Division Method):**Divide the given numbers by prime factors as shown below.

- The product of all the prime factors gives the L.C.M.

\[\therefore \]L.C.M. =2x2x3x3x2= 72

L.C.M. of 36 and 72 =72

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**Highest Common Factor (H.C.F.)**

The highest common factor of the given numbers can be found by the following methods.

**(i) By Listing Factors:** List out the factors of the given numbers and find the highest of the common factors.

Factors of 24 are ** 1, 2,3,4,6,** 8,

Factors of 36 are ** 1,2,3,4,6,**9,

The highest of the common factors is 12.

\[\therefore \]The highest common factor of 24 and 36 is 12,

**(ii) **Division Method: Divide the larger of the given numbers by the smaller one. Subtract and then divide the smaller number by the remainder. Continue until the remainder is 0.The last divisor is the required H.C.F. of the given numbers.

\[\therefore \]H.C.F. of 24 and 36 =12

**(iii) By Prime Factorisation Method (Division Method):**

The product of the common factors =2x 2 x 3=12.

\[\therefore \]H.C.F. of 12 and 24 is 12.

**Note: (i) The H.C.F. of given numbers is not greater than any of the given numbers.**

**(ii) The H.C.F. of two co – primes is 1.**

**(iii) The L.C.M. of given numbers is not less than any of the given numbers. **

**(iv) The L.C.M. of two co – prime is equal to their product.**

**(v) The H.C.F. of two given numbers is always a factor of their L.C.M.**

- Product of two numbers = Product of their H.C.F. and L .C. M.

**e.g.,** Consider the numbers 24 and 36.

L.C.M. of 24 and 36 =72

H.C.F. of 24 and 36 =12

Product of numbers = 24 \[\times \] 36 = 864

Product of H.C.F. and L.C.M. = 12 \[\times \] 72 = 864

\[\therefore \]Product of two numbers = Product of their H.C.F. and L.C.M.

- If two numbers are separately divisible by a number, then their sum and difference are also divisible by that number.

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