Real numbers pt-1

HCF and LCM Calculations HCF (Highest Common Factor): The highest number that divides two or more numbers without leaving a remainder.

• LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.

Problem-solving Examples:

1. Finding HCF and LCM of 105 and 168

   • Using Prime Factorization Method:

     • $105 = 3 imes 5 imes 7$

     • $168 = 2^3 imes 3 imes 7$

   • HCF: $3 imes 7 = 21$

   • LCM: $2^3 imes 3 imes 5 imes 7 = 420$

2. Smallest Number Divisible by 6, 7, 8, 9, and 12 Leaving Remainder 1:

   • Find the LCM of these numbers and then subtract 1 from it.

3. Product and HCF Example:

   • If product of two numbers = 320, HCF = 4:

     • (a) LCM = $rac{320}{4} = 80$

     • (b) Finding Two Numbers: Not always possible without additional information.

4. Finding Pairs of Numbers with Given Sum and HCF:

   • Given sum = 528, HCF = 33:

     • Let the numbers be 33m and 33n. Then $33m + 33n = 528$.

     • $m + n = 16$.

     • Possible pairs: (1,15), (2,14), … (15,1)

5. Identifying Constraints by Product of Two Numbers:

   • If the product is 240, determine which of the following can't be possible as one of the two numbers:

     • (a) CANNOT be 25

     • (b) CANNOT be 8

     • (c) CANNOT be 4. Include reasoning based on prime factorization.

Fundamental Theorem of Arithmetic

• Statement: Every natural number greater than 1 can be uniquely factored into primes.

• This means:

  • If $N$ is a natural number, it can be expressed uniquely as $p<em>1^{e</em>1} imes p<em>2^{e</em>2} imes … imes p<em>n^{e</em>n}$ where $p<em>i$ are prime factors and $e</em>i$ are their respective powers.

• HCF and LCM for Multiple Numbers:

  • HCF(N1, N2, N3) = Product of the smallest powers of each prime factor.

  • LCM(N1, N2, N3) = Product of the largest powers of each prime factor.

Additional Concepts

• Conditions with Prime Factors:

  • If $a$, $b$ are positive integers such that $a = p<em>1^{x</em>1} imes p<em>2^{x</em>2}…$ and $b = p<em>1^{y</em>1} imes p<em>2^{y</em>2}…$, their HCF and LCM can be derived from the powers of the primes.

• Perfect Squares and Factors:

  • If $N$ is a perfect square and has certain factors (like 5 and 6), determine the smallest possible $N$ which fulfills these conditions.

• Proving Irrationality:

  • Prove that a number is irrational under certain conditions or when manipulated.

• Number of Factors Calculation:

  • If a number $N$ is expressed as $p<em>1^{e</em>1} imes p<em>2^{e</em>2}…$, the number of factors of $N$ is given by:

  • $(e<em>1 + 1)(e</em>2 + 1)…$ where $e_n$ are the powers of the prime factors.

Example Problem - Number of Factors

• Find the number of factors of $288$:

  • Prime factorization: $288 = 2^5 imes 3^2$.

  • Number of factors = $(5 + 1)(2 + 1) = 18$.

Special Numbers with Unique Factors

• Example Problem: If a number starts with 3 and has only zeros following it with a total of 50 factors, find the value of the number.

HCF (Highest Common Factor): The highest number dividing two or more numbers without leaving a remainder. LCM (Lowest Common Multiple): The smallest common multiple of two or more numbers.

Examples: 1. For 105 and 168 - HCF: 21, LCM: 420.

2. To find the smallest number divisible by 6, 7, 8, 9, and 12 leaving a remainder of 1, compute the LCM then subtract 1.

3. If the product of two numbers is 320 and HCF is 4, then LCM = $\frac{320}{4} = 80$.

4. For a given sum of 528 with HCF = 33, possible pairs can be formed as multiples of 33.

5. Constraints on the product of two numbers can be deduced via prime factorization.

Fundamental Theorem of Arithmetic: Every natural number > 1 can be uniquely factored into primes. HCF and LCM for multiple numbers are determined via the product of respective prime factors' powers.

Additional Concepts: HCF and LCM derived through prime factor powers, calculations for perfect squares, proving irrationality, and determining the number of factors using prime factorization. Example: For 288 with prime factorization $288 = 2^5 \cdot 3^2$, the number of factors = $(5 + 1)(2 + 1) = 18$. An example case of a number starting with 3 and having specific factors is also mentioned.