least common multiple of 6 and 10
The multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
The multiples of 10 are: 10, 20, 30, 40, 50, 60, ...
The least common multiple (LCM) is the smallest number that is a multiple of both 6 and 10. From the lists above, we can see that the LCM is 30. Therefore, the least common multiple of 6 and 10 is 30.
greatest common factor of 30 and 54
To find the greatest common factor (GCF) of 30 and 54, we need to find all the factors of both numbers and identify the greatest one that they have in common.
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30.
The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54.
The common factors of 30 and 54 are: 1, 2, 3, 6.
The greatest common factor of 30 and 54 is 6. Therefore, the GCF of 30 and 54 is 6.
write 16 + 32 as a product of two factors using the GCF and the distributive property.
16 + 32 = ? (? + ?)
To write 16 + 32 as a product of two factors using the GCF and the distributive property, we need to find the greatest common factor of 16 and 32 and then use the distributive property to express the sum as a product of two factors.
The factors of 16 are 1, 2, 4, 8, 16.
The factors of 32 are 1, 2, 4, 8, 16, 32.
The greatest common factor (GCF) of 16 and 32 is 16, since it is the greatest number that both 16 and 32 are divisible by.
Therefore, we can write 16 + 32 as:
16 + 32 = 16 x 1 + 16 x 2 (using the distributive property)
= 16 x (1 + 2)
So, 16 + 32 can be written as the product of 16 and (1 + 2), or 16(1 + 2). Thus, 16 + 32 = 16(1 + 2).
write 28 + 24 as a product of two factors using the GCF and the distributive property.
28 + 24 = ? (? + ?)
To write 28 + 24 as a product of two factors using the GCF and the distributive property, we need to find the greatest common factor (GCF) of 28 and 24 and then use the distributive property to express the sum as a product of two factors.
The factors of 28 are 1, 2, 4, 7, 14, 28.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
The greatest common factor of 28 and 24 is 4, since it is the greatest number that both 28 and 24 are divisible by.
Therefore, we can write 28 + 24 as:
28 + 24 = 4 x 7 + 4 x 6 (using the distributive property)
= 4 x (7 + 6)
So, 28 + 24 can be written as the product of 4 and (7 + 6), or 4(7 + 6). Thus, 28 + 24 = 4(7 + 6).
caylan is making baked goods for charity bake sale. he places a tray of scones, a tray of muffins and a tray of cookies in the oven. the scones bake for 15 minutes the muffins bake for 12 minutes and the cookies bake for 10 minutes. when one tray is done he removes it and replaces it with another tray of the same item. How many minutes after caylan puts the trays in the oven will he first remove the scones, muffins and the cookies at the same time?
To find the time it takes for Caylan to remove the scones, muffins, and cookies from the oven at the same time, we need to find the least common multiple (LCM) of 15, 12, and 10, which is the smallest multiple that all three numbers share.
First, we need to find the prime factors of each number:
- 15 = 3 x 5
- 12 = 2 x 2 x 3
- 10 = 2 x 5
Then, we need to take the highest power of each prime factor that appears in any of the numbers:
- The highest power of 2 is 2^2 = 4.
- The highest power of 3 is 3^1 = 3.
- The highest power of 5 is 5^1 = 5.
Multiplying these together gives us the LCM:
LCM = 2^2 x 3 x 5 = 60
Therefore, Caylan will first remove the scones, muffins, and cookies at the same time after 60 minutes.
To find the least common multiple (LCM) of two numbers, you can follow these steps:
1. Begin by writing down the prime factorization of each number.
- The prime factorization of 6 is 2 x 3.
- The prime factorization of 10 is 2 x 5.
2. Identify all the prime factors from both numbers.
- In this case, the prime factors are 2, 3, and 5.
3. Take the highest power of each prime factor that appears in both factorizations.
- The highest power of 2 is 2^1.
- The highest power of 3 is 3^1.
- The highest power of 5 is 5^1.
4. Multiply all the highest powers together.
- 2^1 x 3^1 x 5^1 = 2 x 3 x 5 = 30.
Therefore, the least common multiple of 6 and 10 is 30.