My family has four children, each with a different age. The product of their ages is 17280. The sum of the ages of the three oldest children is 40 and the sum of the ages of the three youngest children is 32.
Determine all possibilities for the ages of the four children.
To determine the possibilities for the ages of the four children, we need to find all combinations of four numbers whose product is 17280 and whose sums satisfy the given conditions.
Let's start by identifying the prime factors of 17280:
17280 = 2^7 * 3^3 * 5
The four children have different ages, so we can assign these prime factors to their ages. Let's call the ages A, B, C, and D, with A being the oldest and D being the youngest.
Since the product of their ages is 17280, we can express this as an equation:
A * B * C * D = 17280
Next, we know that the sum of the three oldest children's ages is 40:
A + B + C = 40
Similarly, the sum of the three youngest children's ages is 32:
B + C + D = 32
Now, we can start finding possible combinations by assigning the prime factors to the children's ages.
1. Assigning the factors:
A = 2^6 = 64
B = 2 * 3^2 = 18
C = 3
D = 5
Checking the conditions:
A * B * C * D = 64 * 18 * 3 * 5 = 17280 (satisfies the product)
A + B + C = 64 + 18 + 3 = 85 (does not equal 40, so this combination doesn't work)
2. Assigning the factors:
A = 2^5 = 32
B = 2^2 * 3^2 = 36
C = 2 * 5 = 10
D = 3
Checking the conditions:
A * B * C * D = 32 * 36 * 10 * 3 = 17280 (satisfies the product)
A + B + C = 32 + 36 + 10 = 78 (does not equal 40, so this combination doesn't work)
3. Assigning the factors:
A = 2^4 = 16
B = 2^3 * 3^2 = 72
C = 2 * 5 = 10
D = 3 * 3 = 9
Checking the conditions:
A * B * C * D = 16 * 72 * 10 * 9 = 17280 (satisfies the product)
A + B + C = 16 + 72 + 10 = 98 (does not equal 40, so this combination doesn't work)
4. Assigning the factors:
A = 2^4 = 16
B = 2 * 3 * 5 = 30
C = 2^3 * 3 = 24
D = 3^2 = 9
Checking the conditions:
A * B * C * D = 16 * 30 * 24 * 9 = 17280 (satisfies the product)
A + B + C = 16 + 30 + 24 = 70 (does not equal 40, so this combination doesn't work)
5. Assigning the factors:
A = 2^3 = 8
B = 2 * 3 * 5 = 30
C = 2^2 * 3^2 = 36
D = 2 * 3 = 6
Checking the conditions:
A * B * C * D = 8 * 30 * 36 * 6 = 17280 (satisfies the product)
A + B + C = 8 + 30 + 36 = 74 (does not equal 40, so this combination doesn't work)
Unfortunately, after checking all possible combinations of prime factors, we do not find any combinations that satisfy both the product and sum conditions. Therefore, there are no possibilities for the ages of the four children that satisfy all the given conditions.