Number Systems & Number Theory: JEE Main Math

Number theory in JEE Main appears in permutations and combinations, sequences and series, and directly in divisibility and last-digit problems. While it is not a standalone chapter in the syllabus, JEE Main uses number theory techniques extensively — finding the last digit of a power, testing divisibility, computing GCD, or counting integers satisfying a condition. This guide covers every number theory technique JEE Main has tested.

Divisibility Rules and GCD

Divisibility: n is divisible by 2 if last digit is even; by 3 if sum of digits is divisible by 3; by 4 if last two digits form a number divisible by 4; by 5 if last digit is 0 or 5; by 9 if digit sum divisible by 9; by 11 if alternating digit sum is divisible by 11. GCD (HCF): Euclid's algorithm — gcd(a,b) = gcd(b, a mod b), repeat until remainder is 0. LCM: LCM(a,b) = ab/gcd(a,b). JEE Main uses gcd and lcm in the context of sequences (common terms of two APs) and in permutations (arranging objects in specific patterns).

For sequences: the nth common term of two APs (first AP: a₁, d₁ and second AP: a₂, d₂) is also an AP with common difference = LCM(d₁, d₂). JEE Main asks: which terms are common to the sequences 1, 6, 11, 16, ... and 3, 7, 11, 15, ...? First common term: 11. Common difference of new AP: LCM(5, 4) = 20. So common terms: 11, 31, 51, 71, .... For the sequences chapter, see our Sequences and Series Guide.

Modular Arithmetic and Last Digits

Last digit of a power = last digit of (last digit of base)^power mod 10. The last digit of powers of any digit follows a cycle: 1→1 (cycle 1); 2→2,4,8,6,2,... (cycle 4); 3→3,9,7,1 (cycle 4); 4→4,6,4,6 (cycle 2); 5→5 (cycle 1); 6→6 (cycle 1); 7→7,9,3,1 (cycle 4); 8→8,4,2,6 (cycle 4); 9→9,1,9,1 (cycle 2); 0→0 (cycle 1). To find last digit of 7^(2026): cycle length = 4; 2026 mod 4 = 2; last digit of 7² = 49 → last digit is 9. JEE Main uses this technique regularly in integer-type questions. Take a free mock test on algebra and number theory to practise last-digit and modular arithmetic problems.

Remainder Theorem and Fermat's Little Theorem

Remainder theorem (polynomial): the remainder when P(x) is divided by (x−a) is P(a). JEE Main uses this to find the remainder when a specific polynomial is divided by (x−k) — simply evaluate P(k). For finding remainder of integer division: Wilson's theorem and Fermat's little theorem are rarely needed for JEE Main, but the pattern-identification approach works: find the pattern of remainders mod n as the power increases.

Example: remainder of 3¹⁰⁰ mod 7. Powers of 3 mod 7: 3¹=3, 3²=2, 3³=6, 3⁴=4, 3⁵=5, 3⁶=1, 3⁷=3 (cycle repeats with period 6). 100 = 16×6 + 4, so 3¹⁰⁰ ≡ 3⁴ = 81 ≡ 81 − 11×7 = 81 − 77 = 4 (mod 7). Remainder is 4.

Counting Divisors and Perfect Squares

Number of divisors of n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ: τ(n) = (a₁+1)(a₂+1)...(aₖ+1). Sum of divisors: σ(n) = (p₁^(a₁+1)−1)/(p₁−1) × ... × (pₖ^(aₖ+1)−1)/(pₖ−1). For n to be a perfect square: all exponents must be even. JEE Main uses these in combinatorics-flavoured questions: "In how many ways can 360 be expressed as a product of two factors?" Answer: number of divisors of 360 / 2 (if not perfect square).

Integer Solutions of Equations

Linear Diophantine equations ax + by = c: integer solutions exist if and only if gcd(a,b) | c. If (x₀, y₀) is one solution, the general solution is x = x₀ + (b/gcd)t, y = y₀ − (a/gcd)t for integer t. JEE Main tests counting positive integer solutions to equations with constraints — set up the parametric family and find which values of t give positive solutions.

Exam Strategy

Number theory techniques in JEE Main are tools, not standalone topics — use them when the problem involves "last digit of...", "remainder when...", "how many integers satisfy...", or "which terms of two sequences are common". Recognise the technique needed in 10 seconds: last digit → cycle method; remainder → modular cycle; common terms of APs → LCM. For the permutations and combinations chapter that uses number theory extensively, see our Permutations and Combinations Guide. Upgrade for ₹149/month for number theory and discrete math problems at all JEE difficulty levels.