# FACTORS AND MULTIPLES

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1 FACTORS AND MULTIPLES.(A) Find the prime factors of : (i) (ii) (iii) Ans. (i) (ii) (B.) If P n means prime - factors of n, find : (i) P (ii) P (iii) P (iv) P Ans. (i) F =,,, P (Prime factor of ) = and. (ii) F =,,,,,,, P = and. (iii) F =,,,,,,,, P = and. (iv) F =,,,,,,, P =, and. (C.) List the elements of each of M() and M(). Hence, find the least element in the set M() M(). Is it the L.C.M. of and? Ans. M() = {,,,,,,...} M() = {,,,,,...} Set of common multiples of and = M() M() = {, } The smallest element of this set is. Hence, the LCM of and =. Yes, it is LCM of and. (D). List the elements of each of M() and M(). Hence find the LCM of and. Ans. M() = {,,,,,..} M() = {,,,... } Set of commom multiples of and = M() M() = {, } The smallest element of this set is Hence, the LCM of amd =. List the elements of P and P. Hence, find : (i) P P (ii) P P Ans. We have: = =

2 P = {,, } and P = {, } P (i) (ii) P P P = {,, } {, } = {} = {,, } {, } = {,,, }. (i) If P P = Pn, find the value of n. (ii) If P(n) P() = P(), find the n given that n <. Ans. (i) We have : =, = P = {, } and P = {, } P P = {,} {, } = {} = P Hence, n =. (ii) P(n) P() = P(), P(n) P{,,, } = P{, } P(n) =. Express each one of the following as a product of prime factors : (i) (ii) (iii) (iv) (v) Ans. (i) = (ii) = (iii) = (iv) = (v) =. Find the H.C.F. of (i), (ii), (iii),, (iv),, Ans. (i) (ii) and

3 =, =, =, =, HCF = =. HCF = =. (iii),, =, =, =, HCF = =. (iv),, =, =, =, HCF =.. Find the HCF of the following numbers using prime factorisation method : (i), (ii), (iii), (iv), (v), (vi),, Ans. (i), (ii), =, =, =, =, HCF = =. HCF = =

4 (iii), (iv), =, =, =, =, HCF = =. HCF = =. (v), (vi),, =, =, =, =, HCF = = = HCF = =. Find the HCF of the following numbers using long division method : (i), (ii), (iii), (iv), (v), (vi),, Ans. (i), (ii), HCF of and = HCF of and =

5 (iii), (iv), HCF of and = HCF of and = (v), (vi),, HCF of and =. Now, find the HCF of and HCF of, and =.. Find the greatest number that exactly divides and. Ans. Find the HCF of and The required number =.

6 . Find the greatest number that exactly divides, and. Ans. Now, find the HCF of and The required number =.. Two vessels contain litres and litres of milk. Find the measure of a bucket of maximum capacity which can measure the milk of either vessel an exact number of times : Ans. Required capacity of bucket = litres.. Find the LCM of the following number using prime factorisation method: (i), (ii), (iii), (iv),, (v),, Ans. (i), (ii), = = = = = = = = LCM = = = LCM = = = (iii), (iv),,

7 = = = = = = LCM = = = = = (v),, LCM = = = = = ; = ; = LCM = = =. Find the L.C.M. of the following using common division method: (i),,, (ii),,, (iii),,, (iv),,, (v),,,, (vi),, Ans. (i),,, (ii),,, LCM = = LCM = = (iii),,, (iv),,, LCM = =. LCM = =

8 (v),,,, (vi),, LCM = =. LCM = =. The HCM of two numbers is and their LCM. is. If one of the number is, find the other. Ans. Let the other number be x Product of two numbers = HCF LCM x = x = =. The other number is.. The HCF of two numbers is and their LCM is. If one of number is, find the other. Ans. Let the other number be x. Product of two numbers = HCF LCM x = x = = =. The other number is.. The product of two numbers is and their HCF is. Find their LCM. Ans. HCF LCM = Product of two numbers LCM = LCM = =.. The product of two numbers is and their LCM is. Find their HCF. Ans. HCF LCM = Product of two numbers HCF = LCM = =. The H.C.F. and the L.C.M of two numbers are and respectively. If one of the numbers is, find the other number. Ans. HCF = and LCM =, One number = Product of LCM and HCF = =

9 The other number = Product of LCM and HCF One number = =. The product of two numbers is and their LCM is. Find their HCF. Ans. Product of two numbers = Product of their LCM and HCF Here, product of two number = LCM = HCF = =. Can there be two numbers with HCF and LCM? Give reasons in support of your answer. Ans. No, because HCF of two numbers always divides their LCM.. An electronic device makes a beep after every minutes. Another device makes a beep after every minutes. They beeped together at a.m. At what time will they make the next beep together? Ans. LCM of and = = The next beep will be after minutes i.e. + minutes = a.m.. Six bells commence tolling together and toll at intervals of,,,, and minutes respectively. After what interval of time will they toll together again? Ans. Find the LCM of,,.,,. LCM = =. The six bells toll together again after minutes, i.e. after hours.. Find the least number which when divided by,,, and leaves no remainder. Ans. The least number which is exactly divisible by each given numbers is their LCM. Required number = LCM of,,, and. LCM = least required number

10 ,,,,,,,,,,,, = =,,,,,,,, Hence, the least required number =.. Find the least number which when increased by one is exactly divisible by,,, and.,,,, Ans.,,,,,,,,,,,,,,,, LCM = = The required number = =.

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HCF & LCM Solved Sums 1. The least common multiple of 24, 36, and 40 is A ) 340 b ) 360 c) 230 d) 400 2 24 36 40 => 2 X 2 X 2 X 3 X 1 X 3 X 5 2 12 18 20 => 360 2 6 9 10 3 3 9 5 1 3 5 Ans : 360 2. The LCM

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