CAT Concept of Counting trailing zeroes - Practice Questions & MCQ

Introduction:

The concept of counting trailing zeroes essentially revolves around understanding the number of times 10 is a factor in a given number, especially factorials. Since 10 = 2 x 5, the number of trailing zeroes in a factorial can be determined by the number of pairs of 2 and 5 you can form from its factors.

Application:

Most commonly, this concept is applied to factorials. Given a factorial, one can determine the number of trailing zeros by counting the number of factors of 5 (since factors of 2 are generally more abundant).

Foundation Building Questions:

Question: 

How many trailing zeros are there in 100! (100 factorial)?

Solution: 

To determine the number of trailing zeroes in 100!, we need to find out how many times 10 is a factor. Since 10 = 2 x 5, and there are more factors of 2 than 5, we focus on the number of factors of 5.

Divide 100 by This gives the number of multiples of 5 up to 100.

Next, divide 100 by 25 (since ) This gives the count of numbers up to 100 that have an extra factor of 5.

Therefore, the total count of factors of 5 in 100! is 20 + 4 = 24.

Thus, 100! has 24 trailing zeros.

Tips and Tricks:

1. Focus on 5: Since factors of 2 are more abundant than factors of 5 in factorials, count the number of times 5 can be a factor.

2. Multiple Counts: Numbers like 25, 50, 75, 100, etc., count for more than one factor of 5. Always account for them.

3. Higher Powers: For larger numbers or when examining huge factorials, consider higher powers of 5 such as 125, 625, etc.

Application of Previous Concepts:

Though counting trailing zeroes primarily focuses on factors of 5, the principles of division from the basic Number System (Concept 1) are necessary for quick calculations. Furthermore, recognizing patterns, as mentioned in earlier concepts, is integral to efficiently solve such problems.

While this example serves as an illustrative tool for the concept, students should practise a myriad of problems for holistic comprehension.