Number of Zeroes Concept for Competitive Exams | Trailing Zeroes, Highest Power in Factorial and Solved Examples

Q1. How are zeroes formed at the end of a product?

Zeroes are formed at the end of a product when 2 and 5 are multiplied together since 2 multiplied by 5 equals 10. Every pair of 2 and 5 in the prime factorisation of a product gives one trailing zero. So the number of trailing zeroes in any product equals the minimum of the total count of 2s and 5s in the prime factorisation of that product. Since the power of 2 is always greater than the power of 5 in any factorial, we only need to count the power of 5 to find the number of trailing zeroes. This concept is closely connected to Factors and Divisibility topics.

Q2. How do you find the number of trailing zeroes in a factorial?

To find the number of trailing zeroes in n factorial, find the highest power of 5 in n factorial. Divide n by 5 and take the quotient. Then divide that quotient by 5 again and take the quotient. Keep dividing until the quotient becomes zero. Add all the quotients together — that is the number of trailing zeroes. For example trailing zeroes in 160 factorial = 32 plus 6 plus 1 equals 39. This is one of the most frequently asked questions in factorial based number theory problems.

Q3. How do you find the highest power of a prime number in N factorial?

To find the highest power of a prime number p in n factorial, use successive division. Divide n by p and take the quotient. Divide that quotient by p again. Keep dividing until the quotient is zero. Add all quotients to get the highest power of p in n factorial. For example highest power of 3 in 100 factorial equals 33 plus 11 plus 3 plus 1 equals 48. Both the formula method and successive division method give the same result and are covered in detail on this page with solved examples.

Q4. How do you find the highest power of a composite number in N factorial?

To find the highest power of a composite number in n factorial, first express the composite number as a product of its prime factors. Then find the highest power of each prime factor separately in n factorial. The highest power of the composite number equals the minimum value among the powers obtained after dividing by respective prime factor exponents. For example for 6 equals 2 multiplied by 3, find highest power of 2 and highest power of 3 separately in n factorial, then take the minimum of the two values. Study Factors concept to understand prime factorisation better.

Q5. How do you find the highest power of 1000 in 1000 factorial?

Since 1000 equals 10 raised to 3 equals 2 cubed multiplied by 5 cubed, first find the highest power of 5 in 1000 factorial which equals 200 plus 40 plus 8 plus 1 equals 249. Then divide 249 by 3 since we need 5 cubed for each 1000, giving 83. So the highest power of 1000 in 1000 factorial is 83. This type of problem tests your understanding of both trailing zeroes concept and the Remainder Theorem concepts together.

Q6. Which competitive exams ask number of zeroes questions?

Number of zeroes and trailing zeroes questions are asked in SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB, UPSC CSAT, CAT and all major Indian competitive exams. This topic is also important in AMCAT, eLitmus, TCS NQT and all campus placement aptitude tests. Highest power of a number in factorial questions are particularly common in CAT and SSC CGL. Related topics to strengthen alongside are Factors, Divisibility and Remainder Theorem.

Q7. Why is the power of 5 used instead of power of 2 for trailing zeroes?

In any factorial the power of 2 is always greater than the power of 5. Since every trailing zero requires exactly one pair of 2 and 5, the number of trailing zeroes is limited by whichever factor is smaller. Since 5 always appears less frequently than 2 in factorial, the power of 5 is the limiting factor and therefore determines the number of trailing zeroes. For example in 10 factorial, power of 2 is 8 but power of 5 is only 2, so there are only 2 trailing zeroes in 10 factorial. This is a fundamental property of prime numbers in mathematics.

Q8. How can I practice number of zeroes questions for competitive exams?

After understanding the concept thoroughly you can practice on our Exercise on Number of Zeroes page which contains a large number of solved practice questions covering all types of trailing zeroes and highest power problems asked in SSC CGL, Bank PO, CAT, AMCAT, eLitmus and TCS NQT exams. You may also want to study the related topics of Factors, Remainder Theorem and Unit Digit to strengthen your complete number theory preparation. All questions come with detailed step by step solutions and everything is completely free with no registration required.