# ISEE Upper Level Quantitative : How to find the least common multiple

## Example Questions

### Example Question #1 : How To Find The Least Common Multiple

What is the least common multiple of 15 and 25?

Explanation:

is the lowest number that is a multiple of both 15 and 25, so we see which is the first number that appears in both lists of multiples.

The multiples of 15:

The multiples of 25:

### Example Question #2 : How To Find The Least Common Multiple

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Not enough information is given to answer the question.

Not enough information is given to answer the question.

Explanation:

We show that the given information is not enough by taking two cases:

and

and  divide into , so  and .

is prime and , so

.

Therefore, if , (b) is greater, and if , (a) is greater.

### Example Question #3 : How To Find The Least Common Multiple

Which is the greater quantity?

(a)

(b)

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) and (b) are equal.

Explanation:

The prime factorizations of 50 and 60 are:

The greatest common factor of 50 and 60 is the product of the prime factors they share:

The least common multiple of 50 and 60 is the product of all of the prime factors, with shared factors counted once:

,

(a) and (b) are equal.

Note: it is also a property of the integers that the product of the GCF and the LCM of two integers is equal to the product of the two integers themselves.

### Example Question #4 : How To Find The Least Common Multiple

Which is the greater quantity?

(a)

(b)

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(a) is greater

Explanation:

(a)

(b) To find , list their factors:

To find ,examine their prime factorizations:

(a) is greater.

### Example Question #5 : How To Find The Least Common Multiple

Which of the following is the greater quantity?

(A) The least common multiple of 25 and 30

(B) 300

(B) is greater

It is impossible to determine which is greater from the information given

(A) is greater

(A) and (B) are equal

(B) is greater

Explanation:

To find  we can list some multiples of both numbers and discover the least number in both lists:

, so (B) is greater

### Example Question #6 : How To Find The Least Common Multiple

Which of the following is the least common multiple of 25 and 40?

Explanation:

List the first few multiples of both 25 and 40:

The least number in both lists of factors is 200.

### Example Question #7 : How To Find The Least Common Multiple

Multiply the least common multiple of 504 and 624 by the greatest common factor of 504 and 624.

Explanation:

The product of the least common multiple of any two integers and the greatest common factor of the same two integers is the product of the two integers themselves. Therefore,

.

### Example Question #8 : How To Find The Least Common Multiple

, , , , and are five distinct prime integers. Give the least common multiple of and .

Explanation:

If two integers are broken down into their prime factorizations, their greatest common factor is the product of the prime factors that appear in one or both factorizations.

Since , , , , and are distinct prime integers, the two expressions can be factored into their prime factorizations as follows - with their common prime factors underlined:

The LCM collects each of the factors:

### Example Question #9 : How To Find The Least Common Multiple

Define an operation  as follows:

For all positive integers  and

.

Evaluate .

Explanation:

To find the LCD and GCF of 100 and 80, first, find their prime factorizations:

The GCF of the two is the product of their shared prime factors, so

The LCM is the product of all factors that occur in one or the other factorization, so

### Example Question #10 : How To Find The Least Common Multiple

Which of the following is the least common multiple of  and