CBSE Class 10 Maths Notes Chapter 1 Real Numbers

Product of Two Numbers = HCF X LCM of the Two Numbers

• For any two positive integers a and b, $a\times b=H.C.F\times L.C.M.$
• Example – For 36 and 56, the H.C.F is 4 and the L.C.M is 504 $36\times 56=2016$ $4\times 504=2016$ Thus, $36\times 56=4\times 504$
• Let us consider another example: For 5 and 6, the H.C.F is 1 and the L.C.M is 30 5 × 6 = 30 1 × 30 =30 Thus, 5 × 6 = 1 × 30
• The above relationship, however, doesn’t hold true for 3 or more numbers

Real Numbers for Class 10 Solved Examples

Example 1:

Find the largest number that divides 70 and 125 leaving the remainder 5 and 8 respectively.

Solution:

First, subtract the remainder from the number. (i.e) 70-5 = 65 125-8 = 117. Thus, we need to find the largest number that divides 65 and 117 and leaves the remainder 0. To find the largest number, take the HCF of 65 and 117. Finding HCF of 65 and 117. 65 = 5×13 117 = 3×3×13. Hence, HCF (65, 117) = 13. Therefore, the largest number that divides 70 and 125 leaving the remainder 5 and 8 respectively is 13.

Example 2:

Find the LCM of 306 and 657, given that HCF (306, 657) = 9.

Solution:

Given that, HCF (306, 657) = 9. We know that HCF × LCM = Product of Numbers Hence, 9×LCM = 306×657 9×LCM = 201042 LCM = 201042/9 LCM = 22338. Therefore, LCM of 306 and 657 is 22338.

Example 3:

Prove that 1/√2 is an irrational number.

Solution:

To prove 1/√2 is an irrational number. Now, let us take the opposite assumption. (i.e) Take 1/√2 is a rational number. We know that rational numbers are the numbers that can be written in the form of p/q, where q is not equal to 0. (p and q are two co-prime numbers) Hence, 1/√2 = p/q. Now, simplify the above equation by multiplying √2 on both sides. 1 = (p√2)/q q = p√2 Hence, we get q/p = √2. Here, p and q are integers, and hence q/p is a rational number. But, √2 is an irrational number. Hence, our assumption is wrong. Therefore, 1/√2 is an irrational number. Hence, proved.