respectively involved in the three numbers. LCM ( , , ) = × × = Remark : Notice, × × HCF ( , , ) × LCM ( , , ). So, the product of three numbers is not equal to the product of their HCF and LCM. EXERCISE .
. Express each number as a product of its prime factors: (i) (ii) (iii) (iv) (v) . Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) and (ii) and (iii) and .
Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) , and (ii) , and (iii) , and . Given that HCF ( , ) = , find LCM ( , ). .
Check whether n can end with the digit for any natural number n . . Explain why × × + and × × × × × × + are composite numbers. .
There is a circular path around a sports field. Sonia takes minutes to drive one round of the field, while Ravi takes minutes for the same. Suppose they both start at the * Not from the examination point of view. same point and at the same time, and go in the same direction.
After how many minutes will they meet again at the starting point? . Revisiting Irrational Numbers In Class IX, you were introduced to irrational numbers and many of their properties. You studied about their existence and how the rationals and the irrationals together made up the real numbers.
You even studied how to locate irrationals on the number line. However, we did not prove that they were irrationals. In this section, we will prove that , , and, in general, p is irrational, where p