Number System Notes for CUET UG

Number System is the language of arithmetic. Before a student solves percentages, averages, algebraic simplification, or quantitative aptitude puzzles, the student has to understand how numbers behave. That is why Number System is treated as a foundational topic in almost every aptitude book. In CUET UG, this chapter looks simple on the surface, but it quietly powers many other questions based on divisibility, factors, remainders, and pattern recognition.

Real numbers are all the numbers that can be shown on the number line. This includes positive numbers, negative numbers, fractions, decimals, surds such as $\sqrt{2}$, and constants like $\pi$.

Natural numbers start from 1. Whole numbers include 0. Integers include negative numbers as well. A quick way to remember the hierarchy is to think in layers: natural numbers are inside whole numbers, and whole numbers are inside integers.

This matters in MCQs because examiners often hide small classification traps. If 0 is present, think about whole numbers. If negatives are present, think about integers. If a number has a non-terminating and non-repeating decimal expansion, it is irrational.

Rational numbers can always be written as $p/q$ where $q \ne 0$. That means integers, proper fractions, improper fractions, terminating decimals, and repeating decimals are all rational numbers. For example, 0.125 is rational because it can be written as $1/8$, and $0.\overline{3}$ is rational because it equals $1/3$.

Even and odd behaviour shows up in many fast CUET questions where full calculation is unnecessary. An even number is divisible by 2. An odd number is not divisible by 2. The algebraic forms below are extremely useful because they let us prove parity rules in seconds.

Key parity rules are easy to remember but even more useful when you apply them before calculating. Even + Even = Even, Odd + Odd = Even, Even + Odd = Odd, Even multiplied by anything stays even, and Odd multiplied by Odd stays odd.

Prime and composite numbers create another classic CUET trap. A prime number has exactly two distinct positive factors: 1 and itself. A composite number has more than two positive factors. The number 2 is special because it is the only even prime number. The number 1 is neither prime nor composite because it has only one positive factor.

Students often mark 1 as prime out of habit. That mistake can ruin factor-counting or prime-list questions. Another useful insight is that to test whether a number is prime, it is enough to check divisibility by primes up to its square root.

For example, if Arjun is checking whether 97 is prime, he only needs to test divisibility by 2, 3, 5, and 7 because $\sqrt{97}$ is less than 10. Since none of those primes divides 97, the number is prime.

Fractions are rational numbers written as $p/q$. They are typically classified as proper fractions, improper fractions, and mixed fractions. CUET questions may not ask these names directly, but understanding them helps in approximation and comparison questions.

A proper fraction has numerator smaller than denominator, such as $3/7$. An improper fraction has numerator greater than or equal to denominator, such as $9/4$. A mixed fraction combines an integer and a proper fraction, such as $2\frac{1}{3}$.

Decimal conversions also save time. Students should know the common ones instantly: $1/2 = 0.5$, $1/4 = 0.25$, $3/4 = 0.75$, $1/5 = 0.2$, and $1/8 = 0.125$.

Number line sense is another underestimated skill. On the number line, numbers increase as we move to the right. This becomes crucial when comparing negative numbers. Between -3 and -8, the greater number is -3 because it lies closer to zero and further to the right.

If Priya learns to use the number line mentally, many sign-based questions become immediate and she avoids avoidable mistakes in integer comparison.

A factor of a number divides it exactly. A multiple of a number is obtained by multiplying it with integers. This sounds basic, but factor-multiple confusion is one of the most common student errors in objective exams.

For 12, the factors are 1, 2, 3, 4, 6, and 12. The multiples are 12, 24, 36, 48, and so on. A factor is always less than or equal to the number. A multiple can be equal to or greater than the number. Every number has finitely many factors but infinitely many multiples.

Prime factorisation is the key that unlocks many problems in Number System. Every composite number can be expressed uniquely as a product of prime numbers.

Once prime factorisation is done, factor-counting becomes straightforward. If a number is written as $N = p^a q^b r^c$, then the number of positive factors is $(a+1)(b+1)(c+1)$.

For example, $72 = 2^3 \times 3^2$, so the number of positive factors is $(3+1)(2+1)=12$.

This chapter also supports HCF-LCM style thinking, because once numbers are expressed in prime powers, it becomes much easier to identify common and uncommon prime exponents.

Divisibility rules are among the fastest scoring tools in CUET UG, but they are most powerful when students understand why they work. That understanding prevents panic when the examiner twists a familiar-looking problem.

A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. A number is divisible by 5 if its last digit is 0 or 5. These work because every higher power of 10 is already divisible by 2 and 5, so only the last digit matters.

A number is divisible by 4 if its last two digits form a number divisible by 4, and divisible by 8 if its last three digits form a number divisible by 8. This happens because 100 is divisible by 4 and 1000 is divisible by 8.

Divisibility by 3 and 9 comes from place value and modular thinking. Since $10 \equiv 1 \pmod 3$ and $10 \equiv 1 \pmod 9$, a number behaves like the sum of its digits when tested against 3 or 9.

That is why 564 is divisible by 3 exactly when 15 is divisible by 3.

A number is divisible by 6 if and only if it is divisible by both 2 and 3. The divisibility rule of 11 is based on the identity $10 \equiv -1 \pmod{11}$, which makes powers of 10 alternate between +1 and -1. That is why the difference between alternate digit sums decides divisibility by 11.

Once students understand this, the rule feels logical rather than magical.

Remainders are one of the most exam-relevant parts of Number System. If a number $a$ is divided by $b$, then it can always be written as $a = bq + r$, where the remainder satisfies $0 \le r < b$.

For example, $53 = 7 \times 7 + 4$, so the quotient is 7 and the remainder is 4. A powerful exam shortcut is to reduce each large number first. If $38 \equiv 3 \pmod 5$ and $27 \equiv 2 \pmod 5$, then $(38 \times 27) \equiv 3 \times 2 = 6 \equiv 1 \pmod 5$.

Unit digit questions depend on cyclicity. The last digit of powers repeats in short cycles. The cycle of powers of 2 is 2, 4, 8, 6. The cycle of powers of 3 is 3, 9, 7, 1. The cycle of powers of 7 is 7, 9, 3, 1. The cycle of powers of 9 is 9, 1.

So to find the unit digit of $2^{103}$, divide 103 by 4. The remainder is 3, so take the third term of the cycle. The answer is 8.

Digits 0, 1, 5, and 6 are even easier because any positive power of those numbers keeps the same last digit.

This habit is the difference between a 5-second answer and a 50-second answer.

Students often lose marks in Number System not because the concepts are too advanced, but because they start calculating before identifying the question type. In CUET UG, the first winning move is classification. Ask whether the question is about number type, parity, divisibility, factorisation, remainder, or unit digit.

If it is a classification question, compare definitions. If it is a divisibility question, check the relevant digit rule. If it is about factor count, factorise first. If it is about powers, move into modular thinking or cyclicity. This way, method selection becomes automatic.

A smart preparation sequence is also important. First build comfort with number types, negative numbers, rational vs irrational numbers, and prime vs composite logic. Next, master divisibility rules and factorisation. After that, practice remainders and unit digits. Finally, shift to mixed timed practice.

Students like Priya improve faster when they review their mistakes by category. A wrong answer in Number System usually comes from one of three causes: weak concept, wrong rule choice, or unnecessary long calculation. Labeling the cause is more useful than just reading the correct option.

The right sequence for revision is simple: read one concept, solve a few examples, and then take a short timed test on that concept. That cycle produces better retention than passive reading.

With this approach, Number System stops feeling like a chapter of disconnected tricks and starts feeling like a clean set of patterns that repeat across many exam questions.