Answer Show Hint: For solving the question, we should know the relation between the L.C.M, H.C.F of the two numbers and the two numbers. It is stated that the product of L.C.M and H.C.F is equal to the product of the two numbers. This property is valid only for two numbers. NumericallyLet L.C.M = L and H.C.F = H of two numbers a, b. ThenL $\times $ H = a$\times $b$\to (1)$By applying this property in the above question, we get the required number. Complete step-by-step answer: Let the numbers be a, b. Let the L.C.M and H.C.F of the numbers are l and h respectively.From the property of H.C.F which states that the quotients that we get after dividing the numbers with their H.C.F are coprime numbers, we can write that$\begin{align} & \dfrac{a}{h}=x\text{ ; }\dfrac{b}{h}=y\text{ where x and y are coprime numbers} \\ & a=xh\text{ ; b=yh} \\ \end{align}$Finding the L.C.M using ladder method \[\begin{align} & x\left| \!{\underline {\, xh,\text{yh} \,}} \right. \\ & y\left| \!{\underline {\, h,\text{yh} \,}} \right. \\ & h\left| \!{\underline {\, h,\text{h} \,}} \right. \\ & \text{ }\left| \!{\underline {\, 1,\text{1} \,}} \right. \\ \end{align}\]L.C.M = $(x)\times (y)\times (h)$$\begin{align} & L.C.M\times H.C.F=(x)\times (y)\times (h)\times (h) \\ & =\left( xh \right)\times \left( yh \right) \\ \end{align}$= a * b=Product of the two numbers.$\therefore $L.C.M * H.C.F = a * b.In the given question L.C.M = 108H.C.F = 9a = 36We have to find b. Applying the formulaL.C.M * H.C.F = a * b.108$\times $9 = 36$\times $bDividing by 36 gives$\begin{align} & \dfrac{108\times 9}{36}=b \\ & b=\dfrac{108}{4} \\ & b=27 \\ \end{align}$So, the required number is 27.Note: We can verify the answer by calculating the L.C.M and H.C.F of the numbers 36 and 27. L.C.M is\[\begin{align} & 9\left| \!{\underline {\, \text{36,27} \,}} \right. \\ & 4\left| \!{\underline {\, \text{4,3} \,}} \right. \\ & 3\left| \!{\underline {\, \text{1,3} \,}} \right. \\ & \text{ }\left| \!{\underline {\, 1,\text{1} \,}} \right. \\ \end{align}\]L.C.M = $9\times 4\times 3$ = 108H.C.F is\[\begin{align} & 9\left| \!{\underline {\, \text{36,27} \,}} \right. \\ & \text{ }\left| \!{\underline {\, \text{4,3} \,}} \right. \\ \end{align}\]H.C.F = 9The L.C.M and H.C.F of 36 and 27 are 108 and 9 respectively.Hence verified.
Calculate GCF, GCD and HCF of a set of two or more numbers and see the work using factorization. Enter 2 or more whole numbers separated by commas or spaces. The Greatest Common Factor Calculator solution also works as a solution for finding:
What is the Greatest Common Factor?The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly into all numbers with zero remainder. For example, for the set of numbers 18, 30 and 42 the GCF = 6. Greatest Common Factor of 0Any non zero whole number times 0 equals 0 so it is true that every non zero whole number is a factor of 0. k × 0 = 0 so, 0 ÷ k = 0 for any whole number k. For example, 5 × 0 = 0 so it is true that 0 ÷ 5 = 0. In this example, 5 and 0 are factors of 0. GCF(5,0) = 5 and more generally GCF(k,0) = k for any whole number k. However, GCF(0, 0) is undefined. How to Find the Greatest Common Factor (GCF)There are several ways to find the greatest common factor of numbers. The most efficient method you use depends on how many numbers you have, how large they are and what you will do with the result. FactoringTo find the GCF by factoring, list out all of the factors of each number or find them with a Factors Calculator. The whole number factors are numbers that divide evenly into the number with zero remainder. Given the list of common factors for each number, the GCF is the largest number common to each list.
Prime FactorizationTo find the GCF by prime factorization, list out all of the prime factors of each number or find them with a Prime Factors Calculator. List the prime factors that are common to each of the original numbers. Include the highest number of occurrences of each prime factor that is common to each original number. Multiply these together to get the GCF. You will see that as numbers get larger the prime factorization method may be easier than straight factoring.
Euclid's AlgorithmWhat do you do if you want to find the GCF of more than two very large numbers such as 182664, 154875 and 137688? It's easy if you have a Factoring Calculator or a Prime Factorization Calculator or even the GCF calculator shown above. But if you need to do the factorization by hand it will be a lot of work. How to Find the GCF Using Euclid's Algorithm
For additional information see our Euclid's Algorithm Calculator.
References[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition. New York, NY: CRC Press, 2003 p. 101. [2] Weisstein, Eric W. "Greatest Common Divisor." From MathWorld--A Wolfram Web Resource. Help With Fractions: Finding the Greatest Common Factor. Wikipedia: Euclidean Algorithm. No worries! We‘ve got your back. Try BYJU‘S free classes today! No worries! We‘ve got your back. Try BYJU‘S free classes today! No worries! We‘ve got your back. Try BYJU‘S free classes today! Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses Open in App Suggest Corrections 15 |