Chapter 3 — Playing With Numbers

What is a number? A number is a symbol or a group of symbols used to count, measure or label. Digits are 0–9. A numeral is a way of writing a number (for example, 507).

Place value and face value
Every digit in a number has a face value (the digit itself) and a place value (digit × value of its position). Example: in 4,307, the face value of 4 is 4 and its place value is 4 × 1000 = 4000.

Expanded form and standard form
A number in expanded form is written as the sum of each digit multiplied by its place value. Example: 5,218 = 5×1000 + 2×100 + 1×10 + 8×1. The standard (or usual) form is 5,218.

Types of numbers
- Natural numbers: 1, 2, 3, ... (used for counting).
- Whole numbers: 0, 1, 2, 3, ... (natural numbers plus 0).
- Integers: ..., −2, −1, 0, 1, 2, ... (includes negatives).
- Even numbers: divisible by 2 (0, 2, 4, ...).
- Odd numbers: not divisible by 2 (1, 3, 5, ...).

Number line
A number line is a straight line with evenly spaced marks representing numbers. It shows ordering and distance between numbers, including negative integers.

Factors and multiples
A factor (or divisor) of a number is an integer that divides it exactly. Example: factors of 12 are 1, 2, 3, 4, 6, 12. A multiple of a number is the product of that number and an integer. Example: multiples of 4 are 4, 8, 12, 16, ...

Prime and composite numbers
A prime number has exactly two distinct positive factors: 1 and itself (e.g., 2, 3, 5, 7). 1 is neither prime nor composite. A composite number has more than two positive factors (e.g., 4, 6, 8, 9).

Divisibility rules (simple tests)
Quick checks to see if one number divides another without performing full division (useful for factorisation and solving problems).

HCF (Highest Common Factor) and LCM (Least Common Multiple)
HCF (also called GCD) is the greatest number that divides two or more numbers. LCM is the smallest positive number that is a multiple of two or more numbers. For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b (when a and b are positive).

How these concepts help in real life
- Place value is used in money, telling time and measurements (e.g., ₹254 = 2 hundreds + 5 tens + 4 ones).
- LCM is useful when finding common schedules (bus timings, repeating events).
- HCF is used to divide things into equal groups without leftovers (sharing sweets equally, cutting ropes into largest equal pieces).
- Divisibility tests speed up checking for factors (useful in quick mental calculations).

What is a factor?
A factor (or divisor) of a number is an integer that divides the number exactly with no remainder. If a is a factor of b, then b = a × k for some integer k. Every number has 1 and itself as factors.

What is a multiple?
A multiple of a number is the product of that number and an integer. For example, multiples of 5 are 5, 10, 15, 20, ... A number b is a multiple of a if b = a × k for some integer k.

Finding factors
Methods:

• Factor pairs: check numbers up to √n to find pairs (d, n/d).
• Prime factorization: write the number as product of primes and combine primes to form factors.

Prime and composite
A prime number has exactly two distinct factors: 1 and itself. A composite number has more than two factors.

Common factors and Highest Common Factor (HCF)
Common factors of two or more numbers are factors they share. The greatest of these is the HCF (also called GCD).

Common multiples and Least Common Multiple (LCM)
Common multiples are multiples shared by two or more numbers. The smallest positive common multiple is the LCM.

Finding HCF and LCM using prime factorization
1. Write prime factorization of each number.
2. HCF = product of common prime factors taken with the smallest powers.
3. LCM = product of all primes appearing in any factorization taken with the highest powers.

Useful relation for two numbers
For two positive integers A and B: A × B = HCF(A, B) × LCM(A, B).

Divisibility tests (quick checks)
Examples: divisible by 2 if last digit is even; by 3 if sum of digits is divisible by 3; by 5 if last digit is 0 or 5; by 9 if sum of digits is divisible by 9; by 10 if last digit is 0.

Tips for students
Use factor trees for prime factorization, number lines to visualize multiples, and Venn diagrams to compare prime factors of two numbers. Practice with small numbers to build intuition.

Prime number: A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13...

Composite number: A composite number is a natural number greater than 1 that has more than two positive divisors. Examples: 4, 6, 8, 9, 12, 15...

Special case: 1 is neither prime nor composite because it has only one divisor (1 itself).

Key properties:

• 2 is the only even prime. Every other even number is composite because it is divisible by 2.
• All primes greater than 2 are odd.
• Every composite number can be expressed as a product of prime numbers (prime factorization). This is guaranteed by the Fundamental Theorem of Arithmetic.

How to test if a number is prime (simple method): To check whether n is prime, try dividing n by primes 2, 3, 5, 7, ... up to the largest prime not exceeding sqrt(n). If none divide n evenly, n is prime; otherwise it is composite.

Prime factorization (factor tree method): Break the number down into prime factors by repeated division. Example: 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7, so 84 = 2^2 × 3 × 7.

Uses: Prime and composite classification and prime factorization are used to find HCF/GCF and LCM, solve divisibility problems, simplify fractions, and in many real-life grouping and arrangement problems.

Definition: Two positive integers are called co-prime or relatively prime if their highest common factor (HCF) or greatest common divisor (GCD) is 1. In other words, they have no prime factor in common.

Key ideas and properties:

• If gcd(a, b) = 1, then a and b are co-prime.
• Any two consecutive integers (n and n+1) are always co-prime.
• The number 1 is co-prime with every integer. Zero is co-prime only with 1 because gcd(0, n) = n.
• If two numbers are co-prime, then their LCM = product of the numbers: lcm(a, b) = a × b.
• If a prime number p does not divide b, then p and b are co-prime.

How to check if two numbers are co-prime

1. By prime factorization: Factor both numbers into primes. If they share no common prime factors, they are co-prime. Example: 8 = 2³ and 15 = 3 × 5. No common prime → co-prime.
2. By Euclid's algorithm (GCD method): Repeatedly apply remainder steps until remainder becomes 0; the last non-zero remainder is the gcd. If gcd = 1 → co-prime. Example: gcd(14,15): 15 = 14×1 + 1, 14 = 1×14 + 0 → gcd = 1 → co-prime.

Simple examples: (8, 15), (14, 15), (9, 28), (25, 4), (17, 20). Consecutive pairs like (7, 8) are always co-prime.

Important relation: For any two positive integers a and b,
a × b = gcd(a, b) × lcm(a, b). If gcd(a, b) = 1, then lcm(a, b) = a × b.

Classroom tip: Use small prime lists (2, 3, 5, 7, 11...) to quickly test factor overlaps for numbers up to a few hundred.

What is Prime Factorization?

Prime factorization of a number is expressing the number as a product of prime numbers. Every composite number can be written as a product of primes. This is useful because primes are the "building blocks" of all natural numbers.

Key terms

• Prime number: A number greater than 1 with exactly two factors: 1 and itself (for example, 2, 3, 5, 7).
• Composite number: A number greater than 1 that has more than two factors (for example, 4, 6, 12).
• Prime factors: The prime numbers that multiply together to give the original number.

Methods to find prime factors

• Factor tree method: Start by splitting the number into two factors. Keep splitting any composite factor until all leaves are prime. Multiply the leaf primes to get the original number.
• Division (ladder) method: Divide the number by the smallest possible prime (2, then 3, then 5, etc.) repeatedly until the quotient becomes 1. The primes you used are the prime factors.

Writing in exponential form

If a prime repeats, write it using powers. For example, if the prime 2 appears three times, write 2^3. This gives a compact prime factorization like 360 = 2^3 × 3^2 × 5.

Simple rules and special cases

• 1 has no prime factors (1 is neither prime nor composite).
• Any prime number's prime factorization is the number itself.
• By the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization (ignoring the order of factors).

How prime factorization helps

• Find HCF (GCD): multiply the common primes with the lowest powers.
• Find LCM: multiply all primes that appear in any factor with the highest powers.
• Solve problems about grouping, dividing objects equally, and simplifying fractions.

What is divisibility? A number A is divisible by another number B if A can be divided by B with no remainder. Tests of divisibility are quick rules that tell us whether a number is divisible by another number without doing full division.

Common tests (with brief explanation):

• Divisible by 2: If the last digit is 0, 2, 4, 6 or 8. (Reason: base 10 units place decides evenness.)
• Divisible by 3: If the sum of the digits is divisible by 3. (Reason: 10 ≡ 1 mod 3, so each place value contributes its digit to the remainder.)
• Divisible by 4: If the number formed by the last two digits is divisible by 4. (Reason: 100 is divisible by 4.)
• Divisible by 5: If the last digit is 0 or 5.
• Divisible by 6: If the number is divisible by both 2 and 3.
• Divisible by 8: If the number formed by the last three digits is divisible by 8. (Reason: 1000 is divisible by 8.)
• Divisible by 9: If the sum of the digits is divisible by 9. (Reason: 10 ≡ 1 mod 9.)
• Divisible by 10: If the last digit is 0.
• Divisible by 11 (useful extra): If the alternating sum of digits (sum of digits in odd places minus sum of digits in even places) is divisible by 11 (including 0). (Reason: 10 ≡ -1 mod 11.)

How to apply these: For large numbers, use only the last relevant digits (one, two, or three digits) or compute digit-sums to test divisibility quickly. Sometimes combine tests: e.g., to test for 12 check divisibility by 3 and 4.

Short proofs (intuitive): Write a number as digits: for example, 372 = 3×100 + 7×10 + 2. Since powers of 10 give simple remainders mod 3, 9, 4, 8 or 11, we can reduce to only a few digits or digit sums when checking divisibility.

Tips: When dividing objects into equal groups (sweets, packets, seats), use tests to see if equal sharing is possible without full division. For combined tests (like 6, 12), check smaller prime-power factors.

The Highest Common Factor (HCF), also called Greatest Common Divisor (GCD), of two or more whole numbers is the largest number that exactly divides each of them. HCF answers the question: 'What is the biggest size we can split these numbers into so that all parts are equal and there is no remainder?'

Common methods to find HCF:

• Listing factors: Write all factors of each number and choose the greatest one common to all.
• Prime factorization: Express each number as a product of primes. For each prime common to all numbers, take the lowest power and multiply them. (E.g. if 2 appears as 2^2 in one and 2^3 in another, take 2^2.)
• Division (Euclid’s) method: Repeatedly divide and take remainders: HCF(a, b) = HCF(b, a mod b). Continue until remainder is 0; the nonzero divisor then is the HCF.

Important ideas:

• Two numbers are co-prime if their HCF is 1 (they have no common factor other than 1).
• For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b (when a and b are positive integers).
• To find HCF of more than two numbers, find HCF of two at a time: HCF(a, b, c) = HCF(HCF(a, b), c).

Worked example (36 and 48) using prime factorization:

• 36 = 2^2 × 3^2
• 48 = 2^4 × 3^1
• Common primes: 2 and 3. Take lowest powers: 2^2 and 3^1 → HCF = 2^2 × 3 = 4 × 3 = 12.

What is LCM?
The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of each of them. In other words, it is the first number (other than 0) where the lists of multiples of the numbers meet.

Why LCM is useful?
LCM helps solve problems where events repeat at regular intervals and you want to know when they will happen together again (e.g., bells ringing, traffic lights, timetables).

Methods to find LCM

• Listing multiples: Write multiples of each number and find the smallest common one. (Good for small numbers.)
• Prime factorization: Express each number as a product of primes. For each prime, take the highest power that appears, then multiply those together to get the LCM.
• Using HCF (GCF): For two numbers a and b, use the relation LCM(a, b) = |a × b| / HCF(a, b). This is quick when you can find the HCF (also called GCD).
• For more than two numbers: Use prime factor highest powers for all numbers, or compute pairwise: LCM(a,b,c) = LCM(LCM(a,b), c).

Notes and cautions
- LCM is defined for nonzero integers. Avoid 0 when learning LCM because multiples of 0 are not helpful for finding a meaningful smallest positive common multiple.
- LCM is always positive (for positive integers).

Small worked example (prime factor method)
Find LCM of 12 and 18:
12 = 2² × 3, 18 = 2 × 3² → take highest powers: 2² × 3² = 4 × 9 = 36. So LCM(12, 18) = 36.

Same example (HCF method)
HCF(12, 18) = 6. Then LCM = (12 × 18) / 6 = 216 / 6 = 36.

HCF (Highest Common Factor) of two or more numbers is the greatest number that exactly divides each of them. It is also called GCD (Greatest Common Divisor).

LCM (Least Common Multiple) of two or more numbers is the smallest positive number that is a multiple of each of them.

Finding HCF and LCM by prime factorisation: Write each number as product of prime powers. For each prime:

• HCF: take the prime with the minimum exponent occurring in the numbers.
• LCM: take the prime with the maximum exponent occurring in the numbers.

Key relationship (for two numbers a and b):

If a and b are two positive integers, and H = HCF(a,b), L = LCM(a,b), then

a × b = H × L

Why this is true (sketch): Let prime factorisations be a = ∏ p_i^{a_i} and b = ∏ p_i^{b_i}. For each prime p_i, H takes p_i^{min(a_i,b_i)} and L takes p_i^{max(a_i,b_i)}. Multiplying H and L gives p_i^{min+max} = p_i^{a_i+b_i} which equals the exponent product a_i + b_i so that H×L = a×b.

Important note: The simple product relation a×b = HCF(a,b)×LCM(a,b) holds for two numbers. For more than two numbers, there is no such direct simple product identity; instead use prime factor rules: HCF takes minimum exponents and LCM takes maximum exponents among all numbers.

How it helps (applications):

• Cutting lengths into largest equal pieces (use HCF). Example: cutting ropes into equal longest pieces.
• Finding when repeating events coincide (use LCM). Example: bells ringing every 12 and 18 minutes — LCM gives next time both ring together.
• Arranging objects into equal rows/columns without leftovers (HCF).
• Scheduling repeating tasks, synchronising lights, gears with teeth counts (LCM).

Practical method tips:

1. Use prime factor trees for small numbers.
2. Use the Euclidean algorithm to find HCF quickly for larger pairs; then compute LCM = (a×b)/HCF.
3. For more than two numbers, find HCF and LCM by prime powers: HCF uses minima of exponents; LCM uses maxima.