Question

The greatest integer \( r \) such that \( 30^r \) divides \( 30! \) is:

• A. \( 8 \)
• B. \( \mathbf{7} \)
• C. \( 6 \)
• D. \( 5 \)

Hint:

Use Legendre's formula to find the highest power of a number dividing a factorial.

Answer 1

The Correct Option is B

We want the largest \( r \) such that \( 30^r = 2^r \cdot 3^r \cdot 5^r \) divides \( 30! \). Use Legendre’s formula to find powers of 2, 3, and 5 in \( 30! \): \[ \left\lfloor \frac{30}{2} \right\rfloor + \left\lfloor \frac{30}{4} \right\rfloor + \left\lfloor \frac{30}{8} \right\rfloor + \left\lfloor \frac{30}{16} \right\rfloor = 15 + 7 + 3 + 1 = 26 \] \[ \left\lfloor \frac{30}{3} \right\rfloor + \left\lfloor \frac{30}{9} \right\rfloor + \left\lfloor \frac{30}{27} \right\rfloor = 10 + 3 + 1 = 14 \] \[ \left\lfloor \frac{30}{5} \right\rfloor + \left\lfloor \frac{30}{25} \right\rfloor = 6 + 1 = 7 \] Smallest of these values = \( r = \boxed{7} \)