Number System

Divisibility, factors, HCF, LCM, remainders.

Practice — 3 questions Ask AI

Divisibility rules

Rules for 2, 3, 4, 5, 6, 7, 8, 9, 11.

Number system — natural, rational, real, divisibility, LCM/HCF

Notes

TYPES OF NUMBERS:

Natural numbers (N): 1, 2, 3, ...
Whole numbers (W): 0, 1, 2, 3, ...
Integers (Z): ..., −2, −1, 0, 1, 2, ...
Rational numbers (Q): p/q form, q ≠ 0. Decimals terminating or repeating.
Irrational numbers: π, √2, e. Non-repeating non-terminating decimals.
Real numbers (R): rational + irrational.
Complex numbers (C): a + ib form.

Special subsets:

Prime: divisible only by 1 and itself (2, 3, 5, 7, 11, ...).
Composite: > 1 and not prime.
Even: divisible by 2.
Odd: not divisible by 2.

KEY PROPERTIES:

Closure:

Addition, multiplication closed on N, W, Z, Q, R, C.
Subtraction closed on Z, Q, R, C (not N, W in general).
Division closed on Q{0}, R{0}, C{0}.

Associativity: (a + b) + c = a + (b + c).
Commutativity: a + b = b + a; a × b = b × a.
Distributivity: a × (b + c) = ab + ac.
Identity: 0 (for +); 1 (for ×).
Inverse: −a (for +); 1/a (for ×, a ≠ 0).

DIVISIBILITY RULES (Pack 16 detailed):

By 2: last digit even.
By 3: digit sum div by 3.
By 4: last two digits div by 4.
By 5: last digit 0 or 5.
By 6: div by 2 and 3.
By 7: double last, subtract from rest.
By 8: last three digits div by 8.
By 9: digit sum div by 9.
By 10: last digit 0.
By 11: alternating sum div by 11.

LCM & HCF:

LCM (Least Common Multiple): smallest number divisible by both.
HCF / GCD (Highest Common Factor): largest number that divides both.

Method 1 — Prime factorization:

12 = 2² × 3.
18 = 2 × 3².
HCF: take min powers → 2 × 3 = 6.
LCM: take max powers → 2² × 3² = 36.

Method 2 — Division:

HCF via Euclidean algorithm: gcd(a, b) = gcd(b, a mod b).

Property: a × b = LCM(a, b) × HCF(a, b).

Co-prime numbers: HCF = 1 (e.g., 7 and 12).

REMAINDER THEOREM:

a^n mod m: find pattern.
E.g., 2^10 mod 7? Powers of 2: 2, 4, 1, 2, 4, 1, ... (cycle 3).
- 10 mod 3 = 1. So 2^10 mod 7 = 2.

Fermat's little theorem: a^p ≡ a (mod p) if p prime.

Wilson's theorem: (p−1)! ≡ −1 (mod p) if p prime.

UNIT DIGIT (LAST DIGIT) PROBLEMS:

Cycle of unit digits of powers:

2: 2, 4, 8, 6, 2, 4, 8, 6, ... (cycle 4).
3: 3, 9, 7, 1, ... (cycle 4).
4: 4, 6, 4, 6, ... (cycle 2).
7: 7, 9, 3, 1, ... (cycle 4).
8: 8, 4, 2, 6, ... (cycle 4).
9: 9, 1, 9, 1, ... (cycle 2).
0, 1, 5, 6: cycle 1 (always same).

Q. Unit digit of 7^100?

100 mod 4 = 0 → 4th in cycle of 7 (7, 9, 3, 1) → 1.
(Or 4th position in 0-indexed: position 0 = first... be careful; 100 mod 4 = 0 means we want the LAST in cycle = 1.)

FACTORIAL:

n! = n × (n−1) × (n−2) × ... × 1.
0! = 1.
Trailing zeros in n!: count of factors of 5.
- In 100!: 100/5 + 100/25 + 100/125 = 20 + 4 + 0 = 24 trailing zeros.

SURDS & INDICES:

Surd: irrational expression with root, e.g., √2, √(2+√3).

Properties of indices:

a^m × a^n = a^(m+n).
a^m / a^n = a^(m−n).
(a^m)^n = a^(mn).
a^0 = 1 (a ≠ 0).
a^(-n) = 1/a^n.
a^(1/n) = ⁿ√a.

Properties of surds:

√a × √b = √(ab).
√a / √b = √(a/b).
(√a)² = a.
Rationalizing denominator: × (conjugate).

EXAMPLES:

Q1. Find LCM and HCF of 18 and 24.

18 = 2 × 3². 24 = 2³ × 3.
HCF = 2 × 3 = 6. LCM = 2³ × 3² = 72.

Q2. Number of zeros at end of 100!?

24 (as calculated above).

Q3. Unit digit of 3^25?

Cycle of 3 (3,9,7,1) of length 4. 25 mod 4 = 1. → 1st in cycle = 3.

Q4. Sum of all 2-digit numbers divisible by 7.

14, 21, 28, ..., 98. AP with a=14, l=98, d=7.
n = (98-14)/7 + 1 = 13.
S = 13/2 × (14+98) = 728.

EXAM HOOKS:

Prime check: trial division up to √n.
Trailing zeros in n!: count factors of 5 (Legendre's formula).
Unit digit: cycle pattern.
Co-prime: HCF = 1.
Product of LCM and HCF = product of numbers.
Memorize first 20 primes.

HCF and LCM

Prime factorization, division method, product = HCF × LCM.

HCF, LCM and divisibility rules — the foundations

Notes

Divisibility rules (memorize all):

Divisor	Rule
2	Last digit is even
3	Digit sum divisible by 3
4	Last 2 digits divisible by 4
5	Last digit 0 or 5
6	Divisible by 2 AND 3
7	Double last digit, subtract from rest; repeat. Result divisible by 7?
8	Last 3 digits divisible by 8
9	Digit sum divisible by 9
10	Last digit 0
11	Alternating sum of digits divisible by 11
12	Divisible by 3 AND 4
25	Last 2 digits 00, 25, 50, or 75

HCF (Highest Common Factor) = GCD:

Method 1 (prime factorization): write each as product of primes, take minimum power of each common prime.
Method 2 (Euclidean algorithm — much faster for large numbers):
HCF(a, b) = HCF(b, a mod b), with HCF(a, 0) = a.

Example: HCF(48, 18) = HCF(18, 48 mod 18) = HCF(18, 12) = HCF(12, 6) = HCF(6, 0) = 6.

LCM (Least Common Multiple):

Method 1 (prime factorization): take maximum power of each prime appearing.
Method 2 (using HCF): LCM(a, b) = (a × b) / HCF(a, b).

Example: LCM(48, 18) = (48 × 18) / 6 = 864 / 6 = 144.

Useful identities:

HCF(a, b) × LCM(a, b) = a × b. (Only for TWO numbers; not three+.)
HCF divides every linear combination ax + by.
HCF of fractions: HCF(numerators) / LCM(denominators).
LCM of fractions: LCM(numerators) / HCF(denominators).

Common SSC question types:

"Find smallest number that when divided by a, b, c leaves remainder r each time."
Answer: LCM(a, b, c) + r.

"Find largest number that divides a, b, c leaving same remainder."
Answer: HCF of differences |a−b|, |b−c|, |a−c|.

"Find the number of trailing zeros in n!"
Answer: floor(n/5) + floor(n/25) + floor(n/125) + ... (each 5 in prime factorization contributes one trailing zero; 2s are always abundant).

Example: trailing zeros in 100! = 20 + 4 + 0 = 24.

"Number of factors of n."
If n = p₁^a · p₂^b · p₃^c · ..., then number of factors = (a+1)(b+1)(c+1)...

Example: 72 = 2³ × 3². Factors = (3+1)(2+1) = 12.

Sum of factors: = (p₁^(a+1) − 1)/(p₁ − 1) × similar for p₂, etc.

Number system — natural, rational, real, divisibility, LCM/HCF

Notes

TYPES OF NUMBERS:

Natural numbers (N): 1, 2, 3, ...
Whole numbers (W): 0, 1, 2, 3, ...
Integers (Z): ..., −2, −1, 0, 1, 2, ...
Rational numbers (Q): p/q form, q ≠ 0. Decimals terminating or repeating.
Irrational numbers: π, √2, e. Non-repeating non-terminating decimals.
Real numbers (R): rational + irrational.
Complex numbers (C): a + ib form.

Special subsets:

Prime: divisible only by 1 and itself (2, 3, 5, 7, 11, ...).
Composite: > 1 and not prime.
Even: divisible by 2.
Odd: not divisible by 2.

KEY PROPERTIES:

Closure:

Addition, multiplication closed on N, W, Z, Q, R, C.
Subtraction closed on Z, Q, R, C (not N, W in general).
Division closed on Q{0}, R{0}, C{0}.

DIVISIBILITY RULES (Pack 16 detailed):

LCM & HCF:

LCM (Least Common Multiple): smallest number divisible by both.
HCF / GCD (Highest Common Factor): largest number that divides both.

Method 1 — Prime factorization:

12 = 2² × 3.
18 = 2 × 3².
HCF: take min powers → 2 × 3 = 6.
LCM: take max powers → 2² × 3² = 36.

Method 2 — Division:

HCF via Euclidean algorithm: gcd(a, b) = gcd(b, a mod b).

Property: a × b = LCM(a, b) × HCF(a, b).

Co-prime numbers: HCF = 1 (e.g., 7 and 12).

REMAINDER THEOREM:

a^n mod m: find pattern.
E.g., 2^10 mod 7? Powers of 2: 2, 4, 1, 2, 4, 1, ... (cycle 3).
- 10 mod 3 = 1. So 2^10 mod 7 = 2.

Fermat's little theorem: a^p ≡ a (mod p) if p prime.

Wilson's theorem: (p−1)! ≡ −1 (mod p) if p prime.

UNIT DIGIT (LAST DIGIT) PROBLEMS:

Cycle of unit digits of powers:

2: 2, 4, 8, 6, 2, 4, 8, 6, ... (cycle 4).
3: 3, 9, 7, 1, ... (cycle 4).
4: 4, 6, 4, 6, ... (cycle 2).
7: 7, 9, 3, 1, ... (cycle 4).
8: 8, 4, 2, 6, ... (cycle 4).
9: 9, 1, 9, 1, ... (cycle 2).
0, 1, 5, 6: cycle 1 (always same).

Q. Unit digit of 7^100?

100 mod 4 = 0 → 4th in cycle of 7 (7, 9, 3, 1) → 1.
(Or 4th position in 0-indexed: position 0 = first... be careful; 100 mod 4 = 0 means we want the LAST in cycle = 1.)

FACTORIAL:

n! = n × (n−1) × (n−2) × ... × 1.
0! = 1.
Trailing zeros in n!: count of factors of 5.
- In 100!: 100/5 + 100/25 + 100/125 = 20 + 4 + 0 = 24 trailing zeros.

SURDS & INDICES:

Surd: irrational expression with root, e.g., √2, √(2+√3).

Properties of indices:

a^m × a^n = a^(m+n).
a^m / a^n = a^(m−n).
(a^m)^n = a^(mn).
a^0 = 1 (a ≠ 0).
a^(-n) = 1/a^n.
a^(1/n) = ⁿ√a.

Properties of surds:

√a × √b = √(ab).
√a / √b = √(a/b).
(√a)² = a.
Rationalizing denominator: × (conjugate).

EXAMPLES:

Q1. Find LCM and HCF of 18 and 24.

18 = 2 × 3². 24 = 2³ × 3.
HCF = 2 × 3 = 6. LCM = 2³ × 3² = 72.

Q2. Number of zeros at end of 100!?

24 (as calculated above).

Q3. Unit digit of 3^25?

Cycle of 3 (3,9,7,1) of length 4. 25 mod 4 = 1. → 1st in cycle = 3.

Q4. Sum of all 2-digit numbers divisible by 7.

14, 21, 28, ..., 98. AP with a=14, l=98, d=7.
n = (98-14)/7 + 1 = 13.
S = 13/2 × (14+98) = 728.

EXAM HOOKS:

Prime check: trial division up to √n.
Trailing zeros in n!: count factors of 5 (Legendre's formula).
Unit digit: cycle pattern.
Co-prime: HCF = 1.
Product of LCM and HCF = product of numbers.
Memorize first 20 primes.