Edward has four boxes of marbles. He notices that the product of the number of marbles in box 2,3 and 4 is identical to the number of marbles in box 1. When he adds up the number of marbles in box 2,3 and 4, they add up to 36. The difference in the number of marbles in box 2 and 4 is 1. Find the number of marbles in each of the boxes.
Let's say box 2 has x marbles.
Since the difference in the number of marbles in box 2 and 4 is 1, that means box 4 has x-1 marbles.
Then, the number of marbles in box 3 can be represented as (36 - x - (x-1)).
Simplifying, we get (36 - 2x + 1) = 37 - 2x.
Since the product of the number of marbles in box 2,3 and 4 is identical to the number of marbles in box 1, that means box 1 has (x * (37 - 2x)).
Therefore, we have x * (37 - 2x) = 37 * (x-1) + (x-1).
Expanding the right side of the equation, we get x * (37 - 2x) = 37x - 37 + x - 1.
Combining like terms, we get x^2 - 36x + 36 = 0.
Factoring the equation, we get (x - 6)(x - 6) = 0.
Therefore, x = 6.
So, box 2 has 6 marbles, box 4 has 6-1 = 5 marbles, and box 3 has (36 - 6 - 5) = 25 marbles.
And, box 1 has 6 * 25 = 150 marbles.
Thus, the number of marbles in each of the boxes is as follows:
Box 1: 150 marbles
Box 2: 6 marbles
Box 3: 25 marbles
Box 4: 5 marbles. Answer: \boxed{150, 6, 25, 5}.
A perfect number is an integer greater than zero. When we add up all of the factors less than that number, we get that number. For example, we have the first perfect number 6. The factors of 6 less than itself is 1,2,3. 1+2+3=6. Find another perfect number
The next perfect number after 6 is 28.
The factors of 28 less than itself are 1, 2, 4, 7, and 14.
1 + 2 + 4 + 7 + 14 = 28.
So, 28 is another perfect number.
Let's assume the number of marbles in box 1 is x.
According to the given information, the product of the number of marbles in box 2, 3, and 4 is equal to the number of marbles in box 1. So, we have:
(x) = (number of marbles in box 2) × (number of marbles in box 3) × (number of marbles in box 4) (equation 1)
We are also given that the sum of the number of marbles in box 2, 3, and 4 is 36. So, we have:
(number of marbles in box 2) + (number of marbles in box 3) + (number of marbles in box 4) = 36 (equation 2)
We know that the difference in the number of marbles in box 2 and 4 is 1. So, we have:
(number of marbles in box 2) - (number of marbles in box 4) = 1 (equation 3)
To solve these equations, we can use a method called substitution. We can substitute equation 3 into equation 2 and solve for the variables.
(number of marbles in box 2) + (number of marbles in box 3) + [(number of marbles in box 2) - 1] = 36
Combining like terms:
2(number of marbles in box 2) + (number of marbles in box 3) = 37 (equation 4)
Now we can substitute equation 4 into equation 1:
(x) = (number of marbles in box 2) × (number of marbles in box 3) × [(37 - (number of marbles in box 2))/2]
Simplifying:
2x = (number of marbles in box 2) × (number of marbles in box 3) × (37 - (number of marbles in box 2))
Now we have a quadratic equation that we can solve to find the values of (number of marbles in box 2) and (number of marbles in box 3).
However, since the question only asks for the number of marbles in each box, let's focus on finding the number of marbles in each box.
From equation 3, we know that the difference between the number of marbles in box 2 and 4 is 1. This means (number of marbles in box 4) = (number of marbles in box 2) - 1.
Substituting this into equation 2, we have:
(number of marbles in box 2) + (number of marbles in box 3) + [(number of marbles in box 2) - 1] = 36
Simplifying:
2(number of marbles in box 2) + (number of marbles in box 3) = 37
Using trial and error, we can find that when (number of marbles in box 2) = 18 and (number of marbles in box 3) = 19, the equation holds true.
Finally, we can substitute the values of x, (number of marbles in box 2), (number of marbles in box 3), and (number of marbles in box 4) into equation 1 to find the number of marbles in each box.
(x) = (18) × (19) × (18 - 1) = 3276
Therefore, the number of marbles in each box is as follows:
Box 1: 3276
Box 2: 18
Box 3: 19
Box 4: 17
To find the number of marbles in each box, let's work through the problem step by step:
Let's denote the number of marbles in each box as follows:
Box 1: x
Box 2: y
Box 3: z
Box 4: w
According to the problem, the product of the number of marbles in boxes 2, 3, and 4 is the same as the number of marbles in box 1. This can be expressed as:
y * z * w = x (Equation 1)
We also know that the sum of the number of marbles in boxes 2, 3, and 4 is 36:
y + z + w = 36 (Equation 2)
Additionally, we are given that the difference between the number of marbles in boxes 2 and 4 is 1:
y - w = 1 (Equation 3)
To solve this system of equations, we can use substitution or elimination.
Let's solve by substitution:
From Equation 3, we can rewrite it as:
w = y - 1
Substituting this into Equation 1, we get:
y * z * (y - 1) = x (Equation 4)
Now we can solve for x in terms of y and z:
x = y * z * (y - 1) (Equation 5)
Substituting the value of x into Equation 2, we have:
y + z + (y - 1) = 36
2y + z = 37 (Equation 6)
We now have two equations in terms of y and z (Equations 5 and 6). Let's solve them simultaneously:
First, let's substitute Equation 6 into Equation 5:
x = y * z * (y - 1)
x = y * (37 - 2y) * (y - 1)
x = y(37y - 2y^2 - 37 + 2y)
x = y(2y^2 - 35y + 37)
Now we have a quadratic equation. Since the product of three numbers is positive, we have two cases:
Case 1: All three numbers (x, y, and z) are positive
In this case, y, x, and z must all be positive. Therefore, we can rewrite Equation 5 as follows:
2y^2 - 35y + 37 > 0
However, upon solving this quadratic inequality, we find that there are no positive integer solutions for y, which means this case does not provide a valid solution.
Case 2: Two numbers (y and z) are positive, and one number (x) is negative
In this case, we substitute the value of z in terms of y from Equation 6 into Equation 5:
x = y(37 - 2y) * (y - 1)
x = y(2y^2 - 39y + 37)
Now we need to find values of y that satisfy the following conditions:
1. y is a positive integer.
2. x, y, and z are all positive.
By trying different values, we find that y = 5 satisfies the conditions:
x = 5(2 * 5^2 - 35 * 5 + 37)
x = 5(50 - 175 + 37)
x = 5(50 - 138)
x = 5(-88)
x = -440
Therefore, we have found a valid solution:
Box 1: -440 marbles
Box 2: 5 marbles
Box 3: 4 marbles
Box 4: 4 marbles