Understanding the Age Difference Problem

Let $F$ represent the father's current age and $S$ represent the son's current age, both in complete years.

The problem states that the difference between the squares of their ages is 899:

$F^2 - S^2 = 899$

We are asked to find the father's age when the son was born, which is equal to the difference between their current ages, $F - S$.

Factoring the Equation

We can use the difference of squares algebraic identity: $a^2 - b^2 = (a - b)(a + b)$.

Applying this to the problem:

$ (F - S)(F + S) = 899 $

Let $x = F - S$ (the age difference) and $y = F + S$ (the sum of their ages). The equation becomes:

$ x \cdot y = 899 $

Since $F$ and $S$ represent ages in complete years, they are positive integers. Therefore, $x$ and $y$ must be positive integer factors of 899.

Additionally, since $F > S$, $x = F - S$ must be positive. Also, $y = F + S > F - S = x$. So, we are looking for factor pairs $(x, y)$ of 899 such that $0 < x < y$.

Finding Integer Factors of 899

To find the factors, we can test prime numbers. We need to find the prime factorization of 899.

Testing divisions:

• $899$ is not divisible by 3 (sum of digits is 26).
• $899$ is not divisible by 5.
• Trying larger primes: $899 \div 29 = 31$.

So, the prime factorization is $899 = 29 \times 31$.

The pairs of factors $(x, y)$ of 899, where $x < y$, are:

• Pair 1: $(1, 899)$
• Pair 2: $(29, 31)$

Calculating Possible Age Differences

We also know that $F = \frac{y + x}{2}$ and $S = \frac{y - x}{2}$. For $F$ and $S$ to be integers (representing ages), $x$ and $y$ must have the same parity (both must be odd or both must be even). Since their product $x \cdot y = 899$ is odd, both $x$ and $y$ must be odd.

Both factor pairs satisfy this condition:

Case 1: $x = 1, y = 899$

• $F - S = 1$
• $F + S = 899$
• Adding these gives $2F = 900$, so $F = 450$.
• Subtracting gives $2S = 898$, so $S = 449$.
• This is a valid solution with an age difference of $1$ year.

Case 2: $x = 29, y = 31$

• $F - S = 29$
• $F + S = 31$
• Adding these gives $2F = 60$, so $F = 30$.
• Subtracting gives $2S = 2$, so $S = 1$.
• This is also a valid solution with an age difference of $29$ years.

Determining the Correct Age Difference

The two possible age differences ($F - S$) are 1 year and 29 years.

Comparing these with the given options:

• Option 1: cannot be ascertained
• Option 2: 27 years
• Option 3: 29 years
• Option 4: 31 years

The calculated age difference of 29 years matches Option 3.

Therefore, the age of the father when his son was born is 29 years.