Math Class 10: Prime Factorization and LCM

Number Systems and Algebra

• The prime factorization of a natural number N is 24 × 34 × 53 × 7, and the number of consecutive zeroes in N is dependent on the prime factors.
• The product of two numbers is 3691, and their LCM is 3691, which means their HCF is 1.
• If p and q are two co-prime numbers, then the HCF and LCM of p and q is 1 and pq, respectively.
• The HCF of 52 × 32 and 35 × 53 is 5 × 33.
• If the LCM of 91 and 26 is 182, then the HCF of 91 and 26 is 13.

LCM and HCF

• The HCF and LCM of two numbers are 33 and 264, respectively, and when the first number is divided by 2, the quotient is 33.
• If the LCM and HCF of two non-zero positive numbers are equal, then the numbers must be equal.
• The least number that when divided by 18, 24, 30, and 42 leaves a remainder of 1 is 2519.

Divisibility and Remainders

• The least number of planks required to divide three pieces of timber of lengths 42 m, 49 m, and 63 m is 7.
• The HCF of a number and its remainder when divided by another number is the HCF of the two numbers.

Polynomials

• The graph of a polynomial can be used to determine the number of zeroes.
• The zeroes of a polynomial can be determined graphically, and the number of zeroes is equal to the number of points where the graph of the polynomial intersects the x-axis.
• If one of the zeroes of a quadratic polynomial is –3, then the value of k is dependent on the polynomial.

Quadratic Polynomials

• The sum of the exponents of 2 and 3 in the prime factorization of 3600 is 7.
• The exponent of 2 in the prime factorization of 288 is 5.
• If one of the zeroes of a quadratic polynomial is 0, then the value of c is 0.

Graphs of Quadratic Polynomials

• The graph of a quadratic polynomial can be used to determine the number of zeroes.
• The number of zeroes of a quadratic polynomial is equal to the number of points where the graph of the polynomial intersects the x-axis.
• The graph of a quadratic polynomial can be used to determine the nature of the zeroes.

Linear Equations

• The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is inconsistent.
• The pair of equations y = 0 and y = –7 has no solution.
• A pair of linear equations that has a unique solution x = 2, y = –3 is x + y = –1; 2x – 3y = –5.

Probability

• The probability of an event is a number between 0 and 1.
• The probability of a sure event is 1.
• The probability of an impossible event is 0.
• The probability of an event that is very unlikely to happen is closest to 0.
• The probability of an event that is equally likely to happen or not happen is 0.5.

Experiments and Events

• An experiment is an action or situation that can produce a set of outcomes known as a sample space.
• An event is a set of outcomes of an experiment.
• The probability of an event is the sum of the probabilities of its elementary events.

Solve problems related to prime factorization and LCM in math class 10. Questions include finding consecutive zeroes and determining properties of two numbers given their product and LCM.