john is preparing boxes of school supplies to be sent to countries in need. he packs 124 pencils,100 file folders, and 120 notebooks equally into as many boxes as possible. find the greatest number of boxes that john could pack the items into
To find the greatest number of boxes that John could pack the items into, we need to find the greatest common divisor (GCD) of the numbers 124, 100, and 120.
The prime factorization of 124 is 2*2*31.
The prime factorization of 100 is 2*2*5*5.
The prime factorization of 120 is 2*2*2*3*5.
Taking the highest power of each prime factor that is common to all three numbers, we find that the GCD is 2*2 = 4.
Therefore, John could pack the items into a maximum of 4 boxes.
To find the greatest number of boxes that John could pack the items into, we need to find the greatest common divisor (GCD) of the three quantities: 124 pencils, 100 file folders, and 120 notebooks.
Step 1: Find the GCD of 124 and 100.
We can use the Euclidean algorithm to find the GCD of two numbers:
GCD(124, 100) = GCD(100, 124 % 100) = GCD(100, 24) = GCD(24, 100 % 24) = GCD(24, 4) = GCD(4, 24 % 4) = GCD(4, 0) = 4
Step 2: Find the GCD of the result from step 1 (GCD(4)) and 120.
GCD(4, 120) = GCD(120, 4 % 120) = GCD(120, 4) = GCD(4, 120 % 4) = GCD(4, 0) = 4
Therefore, the greatest number of boxes that John could pack the items into is 4.
To find the greatest number of boxes that John could pack the items into, we need to find the greatest common divisor (GCD) of the three quantities: 124 pencils, 100 file folders, and 120 notebooks.
To find the GCD, we can list the factors of each quantity and identify the common factors:
Factors of 124: 1, 2, 4, 31, 62, 124
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
From the common factors, we can see that the greatest common divisor is 4. This means that John can pack the items into boxes of 4.
To find the number of boxes, we can divide each quantity by the GCD:
Number of pencils per box: 124 pencils / 4 = 31 pencils per box
Number of file folders per box: 100 file folders / 4 = 25 file folders per box
Number of notebooks per box: 120 notebooks / 4 = 30 notebooks per box
Since each box will contain an equal number of each item, the greatest number of boxes that John could pack the items into is determined by the quantity that requires the most items per box. In this case, it is the pencils with 31 pencils per box.
Therefore, John could pack the items into a maximum of 31 boxes.