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Zorluk: OrtaFactors, Multiples, and Prime Factorization

An event organizer needs to pack 144144 apples, 180180 oranges, and 216216 bananas into identical fruit baskets. Each basket must contain only one type of fruit, and all baskets must contain exactly the same number of fruits. What is the minimum total number of fruit baskets the organizer must prepare?

Cevap: 15

Cevap

15
To find the minimum number of baskets, we first need to determine the maximum number of fruits that can be placed in each basket. Since the number of fruits must be the same for all baskets and fruits cannot be mixed, this maximum number is the Highest Common Factor (HCF) of 144, 180, and 216. The HCF is 36. Dividing the total count of each fruit by 36 gives 4 apple baskets, 5 orange baskets, and 6 banana baskets. Adding these together yields a total of 15 baskets.

Adım Adım Çözüm

1
Determine the mathematical operation required to find the maximum number of fruits per basket.
The problem requires finding the Highest Common Factor (HCF) of 144, 180, and 216.
Because the baskets must be identical in capacity, hold only one type of fruit, and we want the minimum number of total baskets (meaning maximum fruits per basket).
2
Perform prime factorization for each fruit quantity.
144 = 2^4 * 3^2; 180 = 2^2 * 3^2 * 5; 216 = 2^3 * 3^3
Prime factorization is the most reliable method for finding the HCF of three large numbers.
3
Calculate the Highest Common Factor (HCF).
HCF = 2^2 * 3^2 = 4 * 9 = 36.
The HCF is the product of the lowest powers of common prime factors present in all three numbers.
4
Calculate the number of baskets required for each individual fruit type.
Apples: 144 / 36 = 4 baskets. Oranges: 180 / 36 = 5 baskets. Bananas: 216 / 36 = 6 baskets.
Dividing the total quantity of each fruit by the maximum capacity of one basket gives the basket count per fruit.
5
Calculate the total number of baskets.
4 + 5 + 6 = 15 baskets.
The question asks for the minimum total number of fruit baskets to be prepared.

Anahtar Kavram

Highest Common Factor (HCF) applied to equitable distribution and grouping
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Konular

Factors, Multiples, and Prime Factorization

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