Class 10 Real Numbers all definitions
Class -10, chapter – 1, Real Numbers
(A) Prime numbers:-
Member of a set of numbers which are divisible only by 1 and itself. In simple words, prime numbers are those numbers which have only two factors 1 and the number itself.
Example: 2, 3, 5, 7 etc.
(B) Composite numbers:-
Composite numbers are basically natural numbers which are divisible by many numbers moreover 1 and itself. Example: 6, 9, 12 etc.
(C) Prime Factorization:
It is the process of splitting a number into its prime factors. Prime factorization method is the best tool to obtain factors of a given number. Normally, prime numbers are used for the division purpose in the process of the prime factorization.
N.B. (Note Book)
(1) HCF of a group of numbers is always less than the LCM of the same group of numbers.
(2) When a number is multiplied with each other n times it is represented as an.
Q. 1. Check whether 6n can end with the digit 0 for any natural number n.
Answer:-
We know that if any natural number ends with 0, it should be divisible by 10, 2 and 5 all.
As of now, 6n (given in the question) can be written as,
6n = (2 × 3)n ————-(1) {6=2×3}
As 2 and 3 are prime factors of 6 in which 5 and 10 are not available.
In this way, we can say that for any value of n, 6n will not be divisible by 5.
Therefore, 6n cannot end with the digit 0 for any natural number n.
Q. 2. Prove that √5 is irrational.
Answer:-
Suppose that √5 is a rational number and is equal to p/q.
p/q = √5 —————(1)
On squaring both sides of equation (1),
(p/q)2 = (√5)2
or, (p2 /q2) = 5
or, p2 = 5q2 —————(2)
From equation (2), it is clear that p2 is divisible by 5.
So, P will also be divisible by 5.
Again, suppose, P=5k where k is an integer.
Thus, based on the equation (2),
p2 = 5q2
or, (5k)2 = 5q2
or, 25 k2 = 5q2
or, q2 = 5 k2 ————–(3)
In the same way, from the equation (3), Q is divisible by 5 which implies that P and Q have common factors.
So this contradicts the fact that P and Q are coprime.
Hence √5 cannot be expressed as p/q or rational number.
In this way, √5 is irrational.
