4.Even and odd numbers:
(I) Even numbers: a counting number divisible by 2 is called an even number.
Thus 0, 2, 4, 6, 8, 10, 12,etc, are all even numbers.
(II) Odd numbers: a counting number not divisible by 2 is called an odd number
Thus, 1, 3, 5, 7, 9, 11, 13, 15 …….. Etc, are all odd numbers
5.Prime numbers: a counting number is called a
prime number if it has exactly two factors, namely itself and 1.
5. EX. All prime numbers less than 100 are:
2, 3, 5, 7, 9, 11, 1, 15, 17, 19, 23, 31, 37, 41, 47, 52, 59, 61, 71, 73, 79, 83,89, 97 test for a number to be prime:
Let p be a given number and let n be the smallest counting number such that n² ≥ p.
Now, test whether p is divisible by any of the prime numbers less than or equal to n.
If yes, the p is not prime otherwise, p is prime.
Ex4, test, which of the following are prime numbers?
(I) 137
(II) 173
(III) 319
(IV) 437
(V) 811
Sol. (I) we know that (12)² > 137.
Prime numbers less than 2 are 2, 3, 5, 7, and 11
173 is a prime number.
(II) We know that (14)²> 319
Prime numbers less than 14 are 2, 3, 5, 7, 11, 13, 17.
Clearly, none of them divides 173.
173 is a prime number.
Prime numbers less than 18 are 2, 3, 5, 7, 11, 13, 17.
Out of these prime numbers.11 divides 319 completely.
319 is not a prime number.
(IV) We know that (21)² > 437
Prime numbers less than 21 are 2,3, 5, 7, 11, 17, 19
Clearly, 437 is divisible by 19.
437 is not a prime number.
(V) We know that (30)² > 811.
Prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Clearly, none of these numbers divides 811.
811 is a prime number.
Composite n umbers: the natural numbers which are not prime are called composite numbers.
6. Two natural numbers a and b are said to be co-prime if their HCF is 1.
Ex. (2, 3), (4,5), (7, 9), (8, 11) etc. are pairs of co-primes.
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