Number Systems are a cornerstone of the CAT Quantitative Aptitude section, testing deep conceptual clarity and logical reasoning. This guided path will take you from the fundamental properties of integers to the advanced techniques required to solve the most challenging CAT problems.
Start with the fundamental types of numbers. This is the basic vocabulary for all Number Systems questions.
All rational and irrational numbers. They can be represented on a continuous number line.
Hover over points to see values.
Dive deeper into integers. Learn what makes them Prime, Composite, Even, Odd, and more with our interactive examples.
Master the essential rules of divisibility. This is a critical time-saving skill for simplifying large numbers in CAT problems.
Learn about prime factorization and how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers.
Every integer > 1 is a unique product of primes.
e.g., 12 = 2² × 3
If N = aᵖ × b^q, Total Factors = (p+1)(q+1).
Go beyond the basics. Learn to calculate the sum and product of factors, and other advanced concepts frequently tested on the CAT.
Calculates Total Factors and Sum of Factors using Geometric Series formula.
Master the algebraic Remainder Formula and explore advanced concepts like cyclicity to solve the toughest CAT remainder problems.
When N is divided by 7, rem is 2. Find rem of 3N divided by 7.
Learn how to find integer solutions for linear equations, a classic advanced problem type that tests your logical framework for number properties.
"A person buys pens for ₹7 each and pencils for ₹5 each. Total spent is ₹95. How many combinations of pens and pencils are possible?"
Isolate a variable: $$ 5y = 95 - 7x $$
We now have to test values for $$x$$
Observe that 95 - 7x should be a multiple of 5. Since 95 is already a multiple of 5, even 7x has to be a multiple of 5. First such instance is when x=5:
$$ 5y = 95 - 35 = 60 $$ (Divisible by 5!)
So, (5, 12) is one solution.
x increases by coeff of y (5).
y decreases by coeff of x (7).
Next x: $$ 5 + 5 = 10 $$
Next y: $$ 12 - 7 = 5 $$
Solution 2: (10, 5)
x increases by coeff of y (5).
y decreases by coeff of x (7).
Next x: $$ 10 + 5 = 15 $$
Next y: $$ 5 - 7 = -2 $$ (Negative!).
Stop here.
Total Solutions: 2
Put your understanding of CAT Number Systems: A Guide to Integer Properties to the test with our premium, adaptive practice questions. Build pacing efficiency and eliminate knowledge gaps now.
Forget formulas for a moment. Learn the 'Systematic Listing' method to build an unshakable intuition for Permutations & Combinations.
How to use 'Extreme Values' to find the range of a sum without calculating every fraction.
Stop calculating! Learn how to use the properties of symmetry to solve hard SD questions in seconds.