Least Common MultipleWorksheets For This SkillMore SkillsFree Online Tests Trial

We call a number M a multiple of another number N if N can divide M. Since 3 can divide 24, 24 is a multiple of 3. Since 4 can also divide 24, 24 is a multiple of 4, too. We call 24 a common multiple of 3 and 4 since 24 is a multiple of both 3 and 4. Are there any other common multiples of 3 and 4 ? Yes, plenty of them. As a matter of fact, any multiple of 24, e.g., 48, 72. etc, is a common multiple of 3 and 4. Another fact is that the product of two numbers is always a common multiple of those two numbers. For example, the product of 3 and 4 is 12, and 12 is definitely a common multiple of 3 and 4.

What number is the least among all the common multiples of 3 and 4 ? The answer is 12. Since none of the numbers less than 12 is a common multiple of 3 and 4, we call 12 the Least Common Multiple(LCM) of 3 and 4.

To find the LCM of two or more numbers, we can use the Prime Factorizations of them. The rule is to pick up all the prime factors that appear in the Prime Factorization of any number, and list each prime factor as many times as it appears in the number where it appears the most times.

Let us use an example to illustrate how to determine the LCM of two numbers by using the Prime Factorization of them.

Example: find the LCM of 24 and 30.

Firstly, we find the Prime Factorization of 24 and 30,
24 = 2 * 2 * 2 * 3
30 = 2 * 3 * 5
It is easy to get all the prime factors that appear in either 24 or 30, or both:
2, 3, 5
Since 2 appears three times in the Prime Factorization of 24, and only once in the Prime Factorization of 30, we list 2 three times
2 * 2 * 2
Since 3 appears once in the Prime Factorization of 24, and also once in the Prime Factorization of 30, we list 3 only once
2 * 2 * 2 * 3
Since 5 appears once in the Prime Factorization of 30, we list 5 only once
2 * 2 * 2 * 3 * 5
That is it. Finally, the product of 2 * 2 * 2 * 3 * 5 is 120, so the LCM of 24 and 30 is 120.
By the way, 120 = 24 * 5 and 120 = 30 * 4 .

Tricks and Tips
If a number M a multiple of another number N, then the LCM of M and N is M itself.

Common DenominatorWorksheets For This SkillMore SkillsFree Online Tests Trial

When you compare or add two fractions which have different denominators, you need to change both fractions to their equivalent fractions with a Common Denominator.

A common denominator for two fractions is nothing but a common multiple of the two denominators. It is up to you which common multiple you want to use. One straight forward way is to use the product of the two denominators as the common denominator. Another way is to use the LCM of the two denominators. In the second way, you are actually using the Lowest Common Denominator(LCD).

Once you have determined the common denominator for two fractions, you can change each fraction to its equivalent fraction by multiplying its numerator by the quotient of dividing the common denominator by its original denominator. This idea is similar to reducing a fraction, where you divide both the numerator and the denominator by the same divisor, and here you multiply both the numerator and the denominator by the same multiplier. In either case, you get an equivalent fraction for the original fraction, in other words, you keep the value of the original fraction unchanged.

Comparing Fractions or Mixed NumbersWorksheets For This SkillMore SkillsFree Online Tests Trial

To compare two fractions, first find out if their denominators are the same. If the denominators are the same, just compare the numerators. The fraction with the bigger numerator is bigger.

If the denominators are different, you need to determine a common denominator for them and change each fraction to its equivalent fraction with that common denominator. Now just compare the numerators. The fraction with the bigger numerator is bigger.

Example: Which one is bigger: or
We know that the LCM of 24 and 30 is 120, so we convert the two fractions to their equivalent fractions with 120 being the common denominator:


Now we compare the new numerators : 25 and 28 . Obviously, 25 is less than 28.
Therefore, is less than , i.e., , is bigger than .

The product of the denominators can always be used as a common denominator. In case you choose that as the common denominator, you need to multiply the first numerator by the second denominator and multiply the second numerator by the first denominator, and then you compare the products:
If numerator1 times denominator2 is bigger than numerator2 times denominator1, then the first fraction is bigger than the second fraction.
If numerator1 times denominator2 is less than numerator2 times denominator1, then the first fraction is less than the second fraction.
If numerator1 times denominator2 is equal to numerator2 times denominator1, then the first fraction is equal to the second fraction.

Example: Which one is bigger: or
Let us do the cross multiplication here:
3 X 30 = 90
7 X 13 = 91
Since 90 is less than 91, is less than , i.e., is bigger than .

When you compare two mixed numbers, first compare their whole numbers. The one with the bigger whole number is bigger. If the whole numbers are the same, then you need to compare their fractions. The one with the bigger fraction is bigger.

Instructions for This AppletFree Online Tests Trial

  1. Click a radio button according to the type of your problem.
  2. Click the check box labeled "Allow Mixed/Whole Numbers" only if you want to add or subtract mixed numbers.
  3. Click the first button to print worksheets; click the second button for the applet to ask you a question; click the third button to input your own question.
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