Question

If 150! is converted to base 7, how many consecutive zeros would be there at the end of it?

Hint:

When finding trailing zeros in a base conversion, count the factors of the base number.

Answer 1

Step 1: Understanding the question.
We are asked how many consecutive zeros will be at the end of 150! when converted to base 7. This is related to the number of factors of 7 in 150!.
Step 2: Finding the number of factors of 7.
To find the number of factors of 7, divide 150 by 7, then by 49, and so on: \[ \left\lfloor \frac{150}{7} \right\rfloor + \left\lfloor \frac{150}{49} \right\rfloor = 21 + 3 = 24 \] Thus, the number of trailing zeros in the base 7 representation of 150! is 9.
Step 3: Conclusion.
The correct answer is \(\boxed{9}\).