Page 41 - Maths Skill - 6

Prime Time 39

(iii) Prime Numbers: A number having only two distinct factors, namely, 1 and the number itself is called a
prime number.
For example, the numbers 2, 3, 5, 7, 11, 13, each Fact-o-meter
has exactly two factors 1 and the number itself, are � Number 2 is the only prime number
prime numbers. which is even.
(iv) Composite Numbers: Numbers having more than � Number 1 is neither a prime nor a
two factors are known as composite numbers. composite number.
For example, the numbers 4, 6, 8, 9, 10, 12, 14, 15, � Any two prime numbers are always co-
16, ... etc. are composite numbers. prime. For example, (2, 3), (3, 5), etc.

(v) Co-prime Numbers: Two numbers are said to be co-prime, if they do not have a common factor other
than 1.

For example, (2, 3), (3, 4), (4, 5), (4, 7), (8, 15) are co-prime numbers.
(vi) Twin-prime Numbers: Two prime numbers are said to be twin-prime numbers, if there is only one
composite number between them.
Or, prime numbers having difference 2.

For example, (3, 5), (5, 7), etc. are twin-prime numbers. (13, 17) is not a twin-prime number because
there are three composite numbers 14, 15, 16 between 13 and 17.

(vii) Prime Triplet: A set of three consecutive prime numbers differing by 2 is called a prime triplet.
For example, the only prime triplet is (3, 5, 7).

(viii) Perfect Number: A number is called a perfect number, if the sum of all its factors, including the number
itself is equal to twice the number.

For example; 6 is a perfect number. Since the factors of 6 are 1, 2, 3, 6, sum of which is 12, which is twice
of 6 that is, 2 × 6 = 12.
Is 28 a perfect number? Find out yourself.

Let’s Attempt

Example 1: Write down all the factors for each of the following.
(i) 12 (ii) 32 (iii) 105
Solution: We know that a factor of a number is an exact divisor of that number, i.e., when it divides a number
the remainder is equal to zero (0).
(i) We know that, 12 = 1 × 12 = 2 × 6 = 3 × 4.
Hence, all the possible numbers which can divide 12 exactly are 1, 2, 3, 4, 6 and 12.
∴ The factors of 12 are 1, 2, 3, 4, 6 and 12.
(ii) We know that,
32 = 1 × 32 = 2 × 16 = 4 × 8.
Hence, all the possible numbers which can divide 32 exactly are 1, 2, 4, 8, 16 and 32.
∴ The factors of 32 are 1, 2, 4, 8, 16 and 32.
(iii) We know that,
105 = 1 × 105 = 3 × 35 = 5 × 21 = 7 × 15.
Hence, all the possible numbers which can divide 105 exactly are 1, 3, 5, 7, 15, 21, 35 and
105.
∴ The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.