# ISEE Upper Level Math : Numbers and Operations

## Example Questions

### Example Question #1 : Least Common Multiple

30 is the least common mutiple of 10 and which of these numbers?

Explanation:

9, 12, and 20 can be eliminated immediately, since none of these have 30 as a multiple [30 divided by any of them yields a remainder].

since 10 is a multiple of 5, so 5 can be eliminated.

The set of multiples of 10 is  and the set of multiples of 15 is ; since 30 is the smallest number that appears on both lists, .

### Example Question #41 : Numbers And Operations

What is the least common multiple of 16 and 20?

Explanation:

is the lowest number that is a multiple of both 16 and 20, so we see which is the first number that appears in both lists of multiples.

The multiples of 16:

The multiples of 20:

### Example Question #3 : Least Common Multiple

Which of the following is the least common multiple of 6, 9, and 12?

648

18

12

36

36

Explanation:

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.

### Example Question #1 : Least Common Multiple

Explanation:

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:

### Example Question #5 : Least Common Multiple

What is the least common multiple of  and ?

Explanation:

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:

### Example Question #42 : Numbers And Operations

Which of the following values of  makes this statement true?

Explanation:

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

Only 375 yields no remainder when divided by 15.

### Example Question #7 : Least Common Multiple

Which of the following values of  makes this statement true?

Explanation:

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

### Example Question #8 : Least Common Multiple

Which of the following values of  makes this statement false?

Explanation:

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

### Example Question #9 : Least Common Multiple

What is the least common multiple of 15 and 18?

Explanation:

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.

### Example Question #10 : Least Common Multiple

What is the least common multiple of 4, 12, and 16?