When a woman is as old as her father is now, she will be five times as old as her son is now. By then, her son will be eight years older than she is now. The combined ages of her father and herself are 100 years. How old is her son?

Let the ages be woman=w, son=s, father=f.

The first sentence f=5s.

Now, the woman will as old as her father in f-w years. The son will then be s+f-w years old. The second sentence says that s+f-w = 8+w.

The 3rd sentence says that w+f = 100.

So, we have three equations, which we can start combining:

5s+w = 100
s+f-w = w+8

5s + w = 100
6s - 2w = 8
-------------
16s = 208

s=13
f = 5s = 65
w = 100-65 = 35

So, in 30 years, the woman will be as old as her father is now(65), and the son will be 43, which is 8 years older than she is now.

The number 34,459,425 is the product of several consecutive positive odd numbers. What is the greatest of these numbers?

To solve this problem, we need to represent the given information in the form of equations. Let's use the variable x to represent the current age of the woman, y to represent the current age of her father, and z to represent the current age of her son.

1) "When a woman is as old as her father is now, she will be five times as old as her son is now." This can be translated into the equation: x + y - x = 5(z - x)

2) "By then, her son will be eight years older than she is now." This can be translated into the equation: z + y = x + 8

3) "The combined ages of her father and herself are 100 years." This means that y + x = 100

Now we have three equations. We can use substitution or elimination to solve the system of equations.

Let's solve using the elimination method:
We have the equations:
x + y - x = 5(z - x) -> y = 5z - 4x
z + y = x + 8
y + x = 100

Substituting the first equation into the second equation, we get:
z + (5z - 4x) = x + 8
6z - 4x = x + 8

Simplifying this equation, we get:
6z - 5x = 8 -> Equation 4

Substituting the third equation into the second equation, we get:
(100 - x) + x = z + 8
100 = z + 8
z = 100 - 8
z = 92 -> Equation 5

Now we have z = 92. We can substitute this value into equation 4 to find x:
6(92) - 5x = 8
552 - 5x = 8
-5x = -544
x = -544 / -5
x = 108.8

Since we are dealing with ages in years, we can round x down to the nearest whole number. Therefore, x = 108.

Finally, we can substitute x = 108 back into equation 3 to find y:
y + 108 = 100
y = 100 - 108
y = -8

Again, since we are dealing with ages in years, we can round y up to the nearest whole number. Therefore, y = -8.

However, the negative age doesn't make sense in this context. So, let's assume there was an error. Let's assume that the combined ages of the woman and her father are 100 years, not their individual ages.

Therefore, if y + x = 100, we can substitute x = 100 - y into equation 4:
6z - 5(100 - y) = 8
6z - 500 + 5y = 8
6z + 5y = 508 -> Equation 6

Substituting x = 100 - y into equation 5, we get:
z = 92

Substituting z = 92 into equation 6, we get:
6(92) + 5y = 508
552 + 5y = 508
5y = 508 - 552
5y = -44
y = -44 / 5
y = -8.8

Again, the negative age doesn't make sense, so let's assume there was an error. Let's assume that the woman's age cannot be greater than her father's age.

Since we want the woman's age to be greater than her father's age, let's assume that her father's age is 50 years. Therefore, y = 50.

Substituting y = 50 into equation 3, we get:
x + 50 = 100
x = 100 - 50
x = 50

Now we have x = 50 and y = 50, which satisfy equation 3.

Substituting x = 50 and y = 50 into equation 2, we get:
z + 50 = 50 + 8
z + 50 = 58
z = 58 - 50
z = 8

Therefore, the son is currently 8 years old.