Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the least square number which is exactly divisible by each of the numbers : $8$ , $12$ , $15$ ,and $20$ .

View step-by-step help

View step-by-step help

Greatest common factor of three or four numbers

Full solution

Q. Find the least square number which is exactly divisible by each of the numbers :

8

,

12

,

15

,and

20

.

Find LCM: To find the least square number that is exactly divisible by $8$ , $12$ , $15$ , and $20$ , we first need to find the least common multiple (LCM) of these numbers. The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.
Prime Factorization: We start by finding the prime factorization of each number. $\newline$ $8 = 2^3$ $\newline$ $12 = 2^2 \times 3$ $\newline$ $15 = 3 \times 5$ $\newline$ $20 = 2^2 \times 5$
Calculate LCM: Next, we take the highest powers of all prime factors found in the factorization of each number to find the LCM. $\newline$ LCM = $2^3 \times 3 \times 5$ (since $2^3$ is the highest power of $2$ , $3$ is the highest power of $3$ , and $5$ is the highest power of $5$ )
Ensure Square Number: Calculating the LCM gives us: $\newline$ LCM = $8 \times 3 \times 5 = 24 \times 5 = 120$
Make LCM Square: Now, we need to ensure that the LCM is a square number. A square number is an integer that is the square of another integer. If the LCM is not a square number, we need to multiply it by the smallest number necessary to make it a square.
Multiply by Factors: The prime factorization of the LCM $120$ is $2^3 \times 3 \times 5$ . To make it a square, we need to have even powers of all prime factors. Currently, the powers of $2$ , $3$ , and $5$ are not even.
Final Result: To make the LCM a square, we need to multiply it by another $2$ (to make the power of $2$ even), by $3$ (to make the power of $3$ even), and by $5$ (to make the power of $5$ even). This will give us the least square number.
Final Result: To make the LCM a square, we need to multiply it by another $2$ (to make the power of $2$ even), by $3$ (to make the power of $3$ even), and by $5$ (to make the power of $5$ even). This will give us the least square number.Multiplying the LCM by $2$ , $3$ , and $5$ gives us: $\newline$ Least square number = LCM $\times 2 \times 3 \times 5 = 120 \times 2 \times 3 \times 5 = 120 \times 30 = 3600$
Final Result: To make the LCM a square, we need to multiply it by another $2$ (to make the power of $2$ even), by $3$ (to make the power of $3$ even), and by $5$ (to make the power of $5$ even). This will give us the least square number.Multiplying the LCM by $2$ , $3$ , and $5$ gives us: $\newline$ Least square number = LCM $\times 2 \times 3 \times 5 = 120 \times 2 \times 3 \times 5 = 120 \times 30 = 3600$ The prime factorization of $2$ $0$ is $2$ $1$ , which is indeed a square number ( $2$ $2$ ). Therefore, $2$ $0$ is the least square number that is exactly divisible by $2$ $4$ , $2$ $5$ , $2$ $6$ , and $2$ $7$ .

More problems from Greatest common factor of three or four numbers

Question

The Huffman Corporation would like to donate

15

20

printers to local schools. The corporation would like to make sure that each school receives the same set of computers and printers, with none left over. What is the greatest number of schools that The Huffman Corporation can donate to?

\newline

\_\_\_\_

Posted 2 months ago

Question

What is the greatest common factor of

2

14

\newline

Posted 2 months ago

Question

What is the least common multiple of

6

12

\newline

Posted 2 months ago

Question

Which numbers are factors of

20

? Select all that apply.

\newline

17

\newline

5

\newline

9

\newline

7

Posted 25 days ago

Question

Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is

12

feet long, and each strand of his Christmas lights will wrap around a

70

-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?

\newline

6

\newline

7

\newline

8

\newline

9

Posted 2 months ago

Question

What is the greatest common factor of

35 y^{4}, 14 y^{4}

63 y^{4}

Posted 4 months ago

Question

What is the greatest common factor of

60 \mathrm{w}, 36 w^{2}

24 w^{4}

Posted 1 month ago

Question

Find the greatest common factor of

75

8

21

Posted 4 months ago

Question

Find the greatest common factor of

110

40

120

Posted 4 months ago

Question

What is the greatest common factor of

6 a, 8 a^{2}

14

Posted 4 months ago