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Factors and Factorials Practice Questions with Detailed Solutions – Level 3

Hard factors and factorials practice on perfect square factors, co-prime pairs, Euler's totient, trailing zeros and the highest power of a number dividing a factorial, with worked solutions.

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Questions, answer key and solutions.

Questions
50
Duration
75 min
Marks
50

Negative marking

Attempts
100
Access
Free
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Normal practice, or a Simulated test

Both use the same questions. Normal practice is untimed and needs no account: choose an option and the answer and solution appear at once. The Simulated test is the real computer-based format, with one 75-minute clock, a question palette and full screen, and your score, the key and the solutions arrive when you submit.

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The computer-based test format: one clock, a question palette, and a score the moment you submit.

  • A clock that does not pause. 75 minutes from the moment you begin, and the paper is submitted for you when it reaches zero.
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About this test

Factors and Factorials Level 3 Practice Quiz: Hard Number System Problems

This Level 3 practice set on Aptitude360 checks whether you can count, sum and classify the factors of large numbers and find prime powers hidden inside factorials. It covers perfect square factors, co-prime factor pairs, Euler's totient, trailing zeros and factorials written in other bases. Every question has a worked solution.

Topics covered in this Factors and Factorials set

  • Counting factors from prime factorisation, including perfect square and perfect cube factors
  • Even and odd factors, and the sum of factors that meet a given condition
  • Factors that are multiples of one number but not divisible by another
  • Writing a number as a product of factor pairs, co-prime pairs or even pairs
  • Euler's totient and counting numbers that are co-prime to a given number
  • Trailing zeros of factorials and of long products, and zeros in another base
  • Highest power of a prime or composite number dividing a factorial or a product
  • Smallest number with a fixed count of factors, and ordered pairs with a given LCM

Key formulas for factors and factorials

  • Number of factors of \( N=p^{a}q^{b}r^{c} \): \( (a+1)(b+1)(c+1) \)
  • Sum of factors of \( N=p^{a}q^{b} \): \( \dfrac{p^{a+1}-1}{p-1}\times\dfrac{q^{b+1}-1}{q-1} \)
  • Highest power of prime \( p \) in \( n! \): \( \sum_{k\ge 1}\left\lfloor \dfrac{n}{p^{k}} \right\rfloor \)
  • Euler's totient, product over all distinct primes \( p \) of \( N \): \( \phi(N)=N\prod\left(1-\dfrac{1}{p}\right) \)
  • Co-prime factor pairs (unordered) of \( N>1 \) with \( k \) distinct primes: \( 2^{k-1} \)

Revision tips for factor and factorial problems

  • Always write the full prime factorisation first; almost every factor question starts from the exponents.
  • For a composite divisor such as twelve, find the power of each prime part separately and take the limiting one.
  • To count factors divisible by a number, pull that number out and count the factors of what remains.
  • When counting unordered factor pairs, check separately whether the number is a perfect square.

Who this factors and factorials set is for

Aspirants for CAT, IPMAT, SSC and banking exams who are comfortable with basic factor counting and want harder Number System practice.

FAQs about Factors and Factorials Practice Questions with Detailed Solutions – Level 3

What is the difference between factors and factorials?
A factor of a number divides it exactly, so the factors of 12 are 1, 2, 3, 4, 6 and 12. A factorial is the product of all natural numbers from 1 up to a given number, so 4 factorial equals 24.
What is 13 factorial?
13 factorial is the product of all natural numbers from 1 to 13, which equals 6,227,020,800.
What are the basic factorial laws?
By definition, 0 factorial equals 1, and n factorial equals n multiplied by the factorial of n minus 1. In basic arithmetic, factorials are defined only for whole numbers.
How do you find the number of trailing zeros in a factorial?
Count the number of times 5 appears as a prime factor, because factors of 2 are always more plentiful. Divide the number by 5, 25, 125 and so on, take the whole-number part each time and add the results.

Test structure

How the paper is split, and how long each part runs. A section you have finished cannot be returned to.

SectionQuestionsTime
Factors and Factorials Level 35075 min

Marking & rules

Total marks
50 across 50 questions.
Negative marking
A wrong answer costs marks. Leaving a question unanswered costs nothing.
Attempts
100 in total. An attempt is spent the moment you begin one.

What you get when you submit

The analysis is ready the moment you submit the paper.

Score and accuracy

Marks, percentage, right, wrong and skipped, and the time you took.

Section by section

The same figures for each section, so a weak one cannot hide in an average.

The full answer key

Every question with your answer, the right one, the working, and the video where there is one.

Where you stand

Your score against the spread of everyone else who has sat this test.

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