Highest power of 72 that can divide 80 factorial 

To find the highest power of \( 72 \) that divides \( 80! \), we start by analyzing \( 72 \) in terms of its prime factors.

### Step 1: Prime Factorization of 72
The number \( 72 \) can be factorized as:
\[
72 = 2^3 \times 3^2
\]

### Step 2: Determine Required Powers of Prime Factors in \( 80! \)
To find the highest power of \( 72 \) that divides \( 80! \), we need to find how many times we can get \( 2^3 \times 3^2 \) as factors in \( 80! \). This involves determining the maximum powers of \( 2 \) and \( 3 \) that divide \( 80! \).

#### Power of 2 in \( 80! \)
The highest power of a prime \( p \) dividing \( n! \) is given by:
\[
\sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor
\]
For \( p = 2 \):
\[
\text{Power of } 2 = \left\lfloor \frac{80}{2} \right\rfloor + \left\lfloor \frac{80}{4} \right\rfloor + \left\lfloor \frac{80}{8} \right\rfloor + \left\lfloor \frac{80}{16} \right\rfloor + \left\lfloor \frac{80}{32} \right\rfloor + \left\lfloor \frac{80}{64} \right\rfloor
\]
Calculating each term:
\[
= 40 + 20 + 10 + 5 + 2 + 1 = 78
\]
Thus, the power of \( 2 \) in \( 80! \) is \( 78 \).

#### Power of 3 in \( 80! \)
For \( p = 3 \):
\[
\text{Power of } 3 = \left\lfloor \frac{80}{3} \right\rfloor + \left\lfloor \frac{80}{9} \right\rfloor + \left\lfloor \frac{80}{27} \right\rfloor + \left\lfloor \frac{80}{81} \right\rfloor
\]
Calculating each term:
\[
= 26 + 8 + 2 + 0 = 36
\]
Thus, the power of \( 3 \) in \( 80! \) is \( 36 \).

### Step 3: Determine the Highest Power of \( 72 \)
To form \( 72 = 2^3 \times 3^2 \), we need powers of \( 2 \) in multiples of 3 and powers of \( 3 \) in multiples of 2.

- From the power of \( 2 \) (which is 78), we can form \( \frac{78}{3} = 26 \) groups of \( 2^3 \).
- From the power of \( 3 \) (which is 36), we can form \( \frac{36}{2} = 18 \) groups of \( 3^2 \).

The limiting factor is \( 18 \), as we need both \( 2^3 \) and \( 3^2 \) to form each \( 72 \).

### Conclusion
The highest power of \( 72 \) that divides \( 80! \) is:
\[
\boxed{18}
\]

Would you like further explanation on any part of this solution?

---

1. What are alternative methods to calculate powers of primes in factorials?
2. How does the formula for power of primes in factorials work in general?
3. Can this method be used for other factorial division problems?
4. Why is \( 3^2 \) the limiting factor in this example?
5. How can this method be extended to analyze larger factorials?

**Tip:** When calculating powers of primes in factorials, ensure you understand the "floor division" method to avoid mistakes in counting prime contributions.

Highest power of 72 that can divide 80 factorial

Learn how to find the highest power of 72 that divides 80 factorial. This detailed solution covers prime factorization, Legendre's formula, and powers of primes in factorials. Suitable for students in Grades 11-12.

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