Least Common Multiple - ISEE Upper Level Quantitative Reasoning

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.

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.

(a)

(b) To find , list their factors:

To find ,examine their prime factorizations:

(a) is greater.

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

, so (B) is greater

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,

.

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:

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

Add:

List the first few multiples of both and

The least number in both lists of factors is .

The least common multiple is the smallest number in value that is a multiple of all three of the numbers. The best way to find the least common multiple (LCM) is to make a quick list of the first few multiples of each number, and then identify the smallest number that is common to all three lists.

Multiples are numbers that you get when multiplying the original number by other numbers.

18 is an answer choice that is common to both 6 and 9, but it is not also a multiple of 12, so it is not correct.

The smallest value that is a common multiple of all three is 36, so this is the LCM.

While 648 is a multiple of all three numbers, it is not the least common multiple of the three numbers.

This equation should be solved from left to right, finding the least common denominator for each pair. First we find the least common denominator of and , which is . The fractions then become and . is then subtracted from that sum:

Find all of the factors of both 9 and 6:

Eliminate all factors that 9 and 6 have in common, in this instance 3 (only eliminate the repeated instance). Multiply all factors that are not in common:

For to be a true statement, must be a multiple of . Each of the four values given is divisible by , as seen below:

For to be true, must be a factor of . Of the five choices, all are factors of except for . This is the correct choice.

For to be a true statement, must be a multiple of 15.

Only 375 yields no remainder when divided by 15.

To find the least common multiple, you need to determine the multiple that both numbers share that is of the least value. List the multiples of each number and identify the first number (least value) that is in both lists:

The LCM of 15 and 18 is 90 since it is the first number that shows up in both lists.

To find the least common multiple, you need to determine the multiple that all three numbers share that is of the least value. List the multiples of each number and identify the first number (least value) that is in both lists:

The LCM of 4, 12, and 16 is 48 since it is the first number that shows up in all three lists.

Find all of the multiples of both numbers:

Find the first number that both sets have in common.

A multiple of x is a number that results when x is multiplied by another whole number.

The least common multiple of two numbers is the smallest number that is a multiple of both the numbers.

24 is a multiple of 8 because 8 times 3 is 24.

24 is also a multiple of 12 because 12 times 2 is 24.

There are no numbers that are smaller than 24 which are also multiples of 8 and 12. Therefore, 24 is the least common multiple.

LCM is the least common multiple of the pair. There are a couple of terms at play here, so first find the LCM of 12 and 48. Since 12 goes into 48, 48 is the LCM. Then, look at xy and x. XY is the LCM between those two. So, multiply those together to get as your answer.