Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said:
"Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$."
How old is the daughter of math teacher?
1. Given the polynomial \( P(x) \) with integer coefficients, we know that \( P(a) = a \) and \( P(0) = p \), where \( a \) is the age of the teacher's daughter and \( p \) is a prime number such that \( p > a \).

2. We use the property that for any polynomial \( P(x) \) with integer coefficients, \( x - y \) divides \( P(x) - P(y) \) for any integers \( x \) and \( y \). Applying this property with \( x = a \) and \( y = 0 \), we get:
   \[
   a - 0 \mid P(a) - P(0)
   \]
   Simplifying, we have:
   \[
   a \mid a - p
   \]

3. Since \( P(a) = a \) and \( P(0) = p \), substituting these values in, we get:
   \[
   a \mid a - p
   \]
   This implies that \( a \) divides \( a - p \). Rewriting \( a - p \) as \( -(p - a) \), we have:
   \[
   a \mid -(p - a)
   \]
   Since \( a \) is a positive integer, we can ignore the negative sign:
   \[
   a \mid (p - a)
   \]

4. Because \( p \) is a prime number and \( p > a \), the only positive integer \( a \) that divides \( p \) and is less than \( p \) is \( 1 \). Therefore, \( a = 1 \).

Conclusion:
\[
\boxed{1}
\]