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Number Properties and Divisibility
Even, Odd, and the Integers
The integers consist of all whole numbers, their negatives, and zero: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$. Within the positive integers, every number is either even numbers or odd numbers.
$$\text{Even}: n = 2k \quad | \quad \text{Odd}: n = 2k + 1 \quad (k \in \mathbb{Z})$$
Even numbers are multiples of 2; odd numbers leave a remainder of 1 when divided by 2.
$k$=Any integer(integer)
$2k$=Generic even number(integer)
$2k + 1$=Generic odd number(integer)
$k = 0$
→$2k = 0$, confirming that 0 is even.
$n = 1$
→1 is the smallest (and only odd) positive integer that is neither prime numbers nor composite numbers.
Addition Rules: Even + Even = Even, Odd + Odd = Even, Even + Odd = Odd.
Multiplication Rules: Even × Any = Even, Odd × Odd = Odd.
Powers: Odd raised to any power stays odd; even raised to any positive power stays even.
Prime Numbers and Factorization
A prime numbers is a positive integer greater than 1 whose only divisibility factors are 1 and itself. Every integer greater than 1 can be uniquely broken down into a product of primes — this is the prime factorization.
$$N = p_1^{a} \times p_2^{b} \times p_3^{c} \times \cdots$$
Any integer $N > 1$ can be expressed as a unique product of prime powers (Fundamental Theorem of Arithmetic).
$p_1, p_2, p_3, \ldots$=Distinct prime factors(prime numbers)
$a, b, c, \ldots$=Exponents (powers) of each prime(non-negative integers)
First Few Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
2 is Special: 2 is the only even prime and the smallest prime. All other primes are odd.
Composite Numbers: Any integer > 1 that is not prime is composite numbers — it has at least three factors.
Counting Factors: If $N = p_1^{a} \times p_2^{b}$, then the total number of factors is $(a+1)(b+1)$.
To find the prime factorization of any number, repeatedly divide by the smallest prime that divides it. The process stops when every factor at the bottom is prime.
$$84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7$$
Breaking 84 into its prime factors: divide by 2 twice, then by 3, leaving 7.
$84$=The number being factorized(unitless)
Division Ladder: Write $N$, divide by 2 as many times as possible, then 3, then 5, 7, 11... until you reach 1.
Shortcut for Large Numbers: Check divisibility by small primes first (2, 3, 5, 7, 11) before moving to larger ones.
Use in LCM/HCF: Once you have prime factorizations, finding LCM and HCF becomes a matter of comparing exponents.
Divisibility Rules
Divisibility rules let you quickly determine whether one number divides another without performing full division. These are essential tools for working with prime factorization, finding HCF and LCM, and solving remainder-based problems.
$$\text{A number } N \text{ is divisible by } d \iff N \mod d = 0$$
A number is divisible by $d$ if it leaves zero remainder when divided by $d$.
$N \mod d$=Remainder when $N$ is divided by $d$(unitless)
By 2: Last digit is even (0, 2, 4, 6, 8).
By 3: Sum of digits is divisible by 3. For $471$: $4 + 7 + 1 = 12$, and 12 is divisible by 3, so 471 is too.
By 4: Last two digits form a number divisible by 4. For $3{,}516$: check $16 \div 4 = 4$ ✓
By 5: Last digit is 0 or 5.
By 6: Must satisfy both rules for 2 and 3 simultaneously.
By 8: Last three digits form a number divisible by 8.
By 9: Sum of digits is divisible by 9 (similar to rule for 3 but stricter).
By 10: Last digit is 0.
By 11: Alternate sum of digits (sum of odd-position digits minus sum of even-position digits) is 0 or divisible by 11. For $2{,}816$: $(2+1)-(8+6) = 3-14 = -11$ ✓
By 12: Must satisfy both rules for 3 and 4 simultaneously.
LCM and HCF
The HCF (also called GCD or Greatest Common Divisor) is the largest number that divides two or more given numbers exactly. It represents the greatest shared prime factorization among the numbers.
$$\text{HCF}(12, 18) = \text{HCF}(2^2 \times 3,\; 2 \times 3^2) = 2^1 \times 3^1 = 6$$
Take the prime factorization of each number and pick the LOWEST power of every common prime.
$\text{HCF}$=Highest Common Factor (Greatest Common Divisor)(unitless)
Prime Factorization Method: Express each number as a product of prime powers. For each common prime, take the minimum exponent.
Division Method: Repeatedly divide by common factors until no common factor remains. The product of all common divisors is the HCF.
HCF of Three or More Numbers: Extend the prime factorization method — take the minimum exponent of each prime that appears in ALL numbers.
Relationship to GCD: HCF = GCD = Greatest Common Divisor. These are all the same concept.
The LCM is the smallest positive number that is a multiple of two or more given numbers. It represents the least value that all the given numbers divide into without leaving a remainder.
$$\text{LCM}(12, 18) = \text{LCM}(2^2 \times 3,\; 2 \times 3^2) = 2^2 \times 3^2 = 36$$
Take the prime factorization of each number and pick the HIGHEST power of every prime that appears in any number.
$\text{LCM}$=Least Common Multiple(unitless)
Prime Factorization Method: For each prime appearing in any factorization, take the maximum exponent across all numbers and multiply.
Division Method: Write numbers in a row, divide by common primes, carry down undivided numbers. Multiply all divisors and remaining numbers.
Use Case — Scheduling: If Event A repeats every 12 days and Event B every 18 days, both coincide every $\text{LCM}(12, 18) = 36$ days.
HCF-LCM Relationship: For two numbers $a$ and $b$, the product $a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$. This also extends to three numbers only in a modified form.
LCM and HCF of Fractions
When dealing with HCF and LCM of fractions, you can reduce the problem to working with the numerators and denominators separately. The formulas are mirror images of each other — HCF swaps what LCM keeps.
$$\text{HCF}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)} \quad | \quad \text{LCM}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$$
For fractions, HCF uses HCF of numerators and LCM of denominators. LCM uses LCM of numerators and HCF of denominators.
$\frac{a}{b}, \frac{c}{d}$=Two fractions in lowest terms(unitless)
Memory Trick: HCF takes the HCF 'upstairs' and LCM 'downstairs'. LCM takes the LCM 'upstairs' and HCF 'downstairs'. It's a cross-swap.
Must Be in Lowest Terms: Convert fractions to their simplest form before applying these formulas, otherwise the results may be incorrect.
Business Application: If two tasks take $\frac{3}{4}$ hour and $\frac{5}{6}$ hour respectively, and they repeat in cycles, the LCM of the fractions tells you when both cycles align.
Remainder Problems and Number Patterns
When a number $N$ is divided by a divisor $d$, it can be expressed in terms of the remainder $r$ and the quotient $q$. Understanding how remainders behave under addition, subtraction, and multiplication is essential for solving competitive-style problems.
$$N = q \times d + r, \quad \text{where } 0 \le r < d$$
Every integer division produces a quotient and a remainder that are uniquely determined.
$N$=The dividend (number being divided)(unitless)
$q$=Quotient (how many times $d$ fits into $N$)(integer)
$r$=Remainder (what's left over)(non-negative integer)
Addition Rule: $(a + b) \mod d = [(a \mod d) + (b \mod d)] \mod d$.
Multiplication Rule: $(a \times b) \mod d = [(a \mod d) \times (b \mod d)] \mod d$.
Negative Remainders: When $r < 0$, add $d$ to make it positive. A remainder of $-2$ mod $7$ is equivalent to $5$ mod $7$.
Power Shortcut: To find $a^n \mod d$, look for a cyclical pattern in $a^1, a^2, a^3, \ldots \mod d$. The pattern always repeats.
Many problems involve identifying repeating patterns in digits, remainders, or unit places of powers. These number series patterns rely on modular arithmetic and cyclical behavior of numbers.
$$\text{Units digit of } 7^n: \quad 7, 9, 3, 1, 7, 9, 3, 1, \ldots \text{ (cycle of 4)}$$
The units digit of powers of any number follows a fixed cycle. For 7, the cycle is 7 → 9 → 3 → 1, repeating every 4 powers.
$n$=The exponent(positive integer)
Units Digit Cycles: 2 and 8 have cycle 4; 3 and 7 have cycle 4; 4 and 9 have cycle 2; 5 and 6 always give 5 and 6; 0 always gives 0.
Finding Position in Cycle: To find the units digit of $a^n$, compute $n \mod \text{cycle length}$, then look up that position in the cycle.
Sum of Digits Patterns: Numbers divisible by 9 have digit sums that are multiples of 9. The remainder of any number divided by 9 equals the remainder of its digit sum divided by 9.