Question

Write the first five multiples of 10 and 20. Find their common multiples and find their LCM.

Answer 1

Hint: First, find the consecutive multiples of each 10 and 20, by multiplying each by 1,2,3,4, and 5. To determine the LCM of the numbers, express the number in terms of the product of its prime factors and multiply all the prime factors the maximum number of times they occur in either number. This is the method of prime factorization.

Complete step-by-step solution -
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated until we are left with 1 as the quotient.
For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now starting with finding the factors of multiples of 10. The smallest multiple of 10 is 1 multiplied by 10, i.e., 10. The other consecutive multiples of 10 can be found by multiplying 10 by 2, 3, 4, and 5, respectively. So, the first five multiples of 10 are: {10,20,30,40,50}.
Now we will find the factors of multiples of 20. The smallest multiple of 20 is 1 multiplied by 20, i.e., 20. The other consecutive multiples of 20 can be found by multiplying 20 by 2, 3, 4, and 5, respectively. So, the first five multiples of 20 are: {20,40,60,80,100}.
So, looking at the above results, we can easily figure out that the common multiples of 10 and 20 among the ones mentioned above are: {20,40}
Now we will find the LCM of 40 and 20. So, let us start by finding the factors of 20. 20 is also an even number, so it can be written as 20 as $20=10\times 2$. Also, we can further break 10 as the product of 2 and 5. Therefore, 20 can be represented as $20=2\times 2\times 5$ .
Now we will find the factors of 40. We know 40 is an even number so it can be written as $40=2\times 20$. Also, we found the factors of 20 in the above part. So, we can say $40=2\times 2\times 2\times 5$ .
Now to find the LCM, we need to multiply all the prime factors the maximum number of times they occur in either number. So, LCM(20,40) is a product of three 2s and one 5, as 2 is occurring 3 times in 40 and 5 is occurring a maximum of one time.
$LCM\left( 20,40 \right)=2\times 2\times 2\times 5=40.$
Therefore, we can conclude that the LCM of 20 and 40 is 40.

Note: Be careful while finding the prime factors of each number. Also, it is prescribed that you learn the division method of finding the LCM as well, as it might be helpful. If in case you are asked to find the LCM of two fractions you must use the formula $LCM=\dfrac{LCM\text{ of the numerator of the fractions}}{HCF\text{ of the denominator of the fractions}}$.