Cole has 20 math books and 16 science books. If he wants to distribute them evenly among some bookshelves so that each bookshelf has the same combination of math and science books, with no books left over, what is the greatest number of bookshelves Cole can use?
bookshelves
To find the greatest number of bookshelves Cole can use, we need to find the greatest common divisor (GCD) of the number of math books and science books.
First, let's find the prime factors of 20:
20 = 2 * 2 * 5
Now, let's find the prime factors of 16:
16 = 2 * 2 * 2 * 2
To find the GCD, we take the minimum exponent for each prime factor:
GCD = 2^2 = 4
Therefore, Cole can use a maximum of 4 bookshelves.
Adele has collected 15 U.S. stamps and 12 international stamps. She wants to display them in identical groups of U.S. and international stamps, with no stamps left over. What is the greatest number of groups Adele can display them in?
groups
To find the greatest number of groups Adele can display the stamps in, we need to find the greatest common divisor (GCD) of the number of U.S. stamps and international stamps.
First, let's find the prime factors of 15:
15 = 3 * 5
Now, let's find the prime factors of 12:
12 = 2 * 2 * 3
To find the GCD, we take the minimum exponent for each prime factor:
GCD = 2^0 * 3^1 * 5^0 = 1 * 3 * 1 = 3
Therefore, Adele can display the stamps in a maximum of 3 groups.
At a school event, adults are seated in groups of exactly 4 and children are seated in groups of exactly 8. If there are the same number of adults as children, what is the minimum number of adults attending?
adults
To find the minimum number of adults attending, we need to find the least common multiple (LCM) of the group size for adults (4) and the group size for children (8).
The LCM is the smallest number that is divisible by both 4 and 8.
The prime factors of 4 are 2 * 2 and the prime factors of 8 are 2 * 2 * 2.
To find the LCM, we take the maximum exponent for each prime factor:
LCM = 2^3 * 2^2 = 8 * 4 = 32.
Therefore, the minimum number of adults attending is 32.
Emmy is creating party favors with pencils and stickers. She has 20 pencils and 16 stickers, and wants each party favor to include the same combination of pencils and stickers, with none left over. What is the greatest number of party favors she can make?
[___] party favors
To find the greatest number of party favors Emmy can make, we need to find the greatest common divisor (GCD) of the number of pencils and stickers.
First, let's find the prime factors of 20:
20 = 2 * 2 * 5
Now, let's find the prime factors of 16:
16 = 2 * 2 * 2 * 2
To find the GCD, we take the minimum exponent for each prime factor:
GCD = 2^2 = 4.
Therefore, Emmy can make a maximum of 4 party favors.
Keenan and Alice make money by walking neighborhood pets. During a conversation about their work, they realize that they are both walking the same total number of pets, even though Keenan walks pets in groups of 4 and Alice walks pets in groups of 8. What is the smallest total number of pets each could be walking?
pets
To find the smallest total number of pets each could be walking, we need to find the least common multiple (LCM) of the group size for Keenan (4) and the group size for Alice (8).
The LCM is the smallest number that is divisible by both 4 and 8.
The prime factors of 4 are 2 * 2 and the prime factors of 8 are 2 * 2 * 2.
To find the LCM, we take the maximum exponent for each prime factor:
LCM = 2^3 * 2^2 = 8 * 4 = 32.
Therefore, the smallest total number of pets each could be walking is 32.
Christina has two pieces of rope, one 16 feet long and the other 12 feet long. For a project, she needs to cut them up to produce many pieces of rope that are all of the same length, with no rope left over. What is the greatest length, in feet, that she can make them?
[___]feet