What is the LCM of 7/10 and 70?

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What is the Least Common Multiple (LCM)?

In mathematics, the least common multiple, also known as the lowest common multiple of two (or more) integers a and b, is the smallest positive integer that is divisible by both. It is commonly denoted as LCM(a, b).

Brute Force Method

There are multiple ways to find a least common multiple. The most basic is simply using a "brute force" method that lists out each integer's multiples.

As can be seen, this method can be fairly tedious, and is far from ideal.

Prime Factorization Method

A more systematic way to find the LCM of some given integers is to use prime factorization. Prime factorization involves breaking down each of the numbers being compared into its product of prime numbers. The LCM is then determined by multiplying the highest power of each prime number together. Note that computing the LCM this way, while more efficient than using the "brute force" method, is still limited to smaller numbers. Refer to the example below for clarification on how to use prime factorization to determine the LCM:

Greatest Common Divisor Method

A third viable method for finding the LCM of some given integers is using the greatest common divisor. This is also frequently referred to as the greatest common factor (GCF), among other names. Refer to the link for details on how to determine the greatest common divisor. Given LCM(a, b), the procedure for finding the LCM using GCF is to divide the product of the numbers a and b by their GCF, i.e. (a × b)/GCF(a,b). When trying to determine the LCM of more than two numbers, for example LCM(a, b, c) find the LCM of a and b where the result will be q. Then find the LCM of c and q. The result will be the LCM of all three numbers. Using the previous example:

Note that it is not important which LCM is calculated first as long as all the numbers are used, and the method is followed accurately. Depending on the particular situation, each method has its own merits, and the user can decide which method to pursue at their own discretion.

LCM of 7 and 10 is the smallest number among all common multiples of 7 and 10. The first few multiples of 7 and 10 are (7, 14, 21, 28, 35, 42, 49, . . . ) and (10, 20, 30, 40, 50, 60, 70, . . . ) respectively. There are 3 commonly used methods to find LCM of 7 and 10 - by listing multiples, by division method, and by prime factorization.

What is the LCM of 7 and 10?

Answer: LCM of 7 and 10 is 70.

Explanation:

The LCM of two non-zero integers, x(7) and y(10), is the smallest positive integer m(70) that is divisible by both x(7) and y(10) without any remainder.

Methods to Find LCM of 7 and 10

The methods to find the LCM of 7 and 10 are explained below.

• By Prime Factorization Method
• By Division Method
• By Listing Multiples

LCM of 7 and 10 by Prime Factorization

Prime factorization of 7 and 10 is (7) = 71 and (2 × 5) = 21 × 51 respectively. LCM of 7 and 10 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 21 × 51 × 71 = 70.
Hence, the LCM of 7 and 10 by prime factorization is 70.

LCM of 7 and 10 by Division Method

To calculate the LCM of 7 and 10 by the division method, we will divide the numbers(7, 10) by their prime factors (preferably common). The product of these divisors gives the LCM of 7 and 10.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 7 and 10. Write this prime number(2) on the left of the given numbers(7 and 10), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (7, 10) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
• Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 7 and 10 is the product of all prime numbers on the left, i.e. LCM(7, 10) by division method = 2 × 5 × 7 = 70.

LCM of 7 and 10 by Listing Multiples

To calculate the LCM of 7 and 10 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 7 (7, 14, 21, 28, 35, 42, 49, . . . ) and 10 (10, 20, 30, 40, 50, 60, 70, . . . . )
• Step 2: The common multiples from the multiples of 7 and 10 are 70, 140, . . .
• Step 3: The smallest common multiple of 7 and 10 is 70.

∴ The least common multiple of 7 and 10 = 70.

☛ Also Check:

1. Example 1: Verify the relationship between GCF and LCM of 7 and 10.

   Solution:

   The relation between GCF and LCM of 7 and 10 is given as, LCM(7, 10) × GCF(7, 10) = Product of 7, 10

   Prime factorization of 7 and 10 is given as, 7 = (7) = 71 and 10 = (2 × 5) = 21 × 51

   LCM(7, 10) = 70 GCF(7, 10) = 1 LHS = LCM(7, 10) × GCF(7, 10) = 70 × 1 = 70 RHS = Product of 7, 10 = 7 × 10 = 70 ⇒ LHS = RHS = 70

   Hence, verified.

Show Solution >

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The LCM of 7 and 10 is 70. To find the LCM (least common multiple) of 7 and 10, we need to find the multiples of 7 and 10 (multiples of 7 = 7, 14, 21, 28 . . . . 70; multiples of 10 = 10, 20, 30, 40 . . . . 70) and choose the smallest multiple that is exactly divisible by 7 and 10, i.e., 70.

How to Find the LCM of 7 and 10 by Prime Factorization?

To find the LCM of 7 and 10 using prime factorization, we will find the prime factors, (7 = 7) and (10 = 2 × 5). LCM of 7 and 10 is the product of prime factors raised to their respective highest exponent among the numbers 7 and 10.
⇒ LCM of 7, 10 = 21 × 51 × 71 = 70.

Which of the following is the LCM of 7 and 10? 10, 70, 3, 11

The value of LCM of 7, 10 is the smallest common multiple of 7 and 10. The number satisfying the given condition is 70.

What is the Relation Between GCF and LCM of 7, 10?

The following equation can be used to express the relation between GCF and LCM of 7 and 10, i.e. GCF × LCM = 7 × 10.

If the LCM of 10 and 7 is 70, Find its GCF.

LCM(10, 7) × GCF(10, 7) = 10 × 7 Since the LCM of 10 and 7 = 70 ⇒ 70 × GCF(10, 7) = 70

Therefore, the GCF (greatest common factor) = 70/70 = 1.

LCM(10, 7) = 2×5×7 = 70

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