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Finding Common Ground – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash II, tailored for Class 7 in Mathematics.
This one-pager compiles key formulas and equations from the Finding Common Ground chapter of Ganita Prakash II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
HCF(a, b) = Highest Common Factor of a and b
HCF is the greatest number that divides both a and b without leaving a remainder. It is useful for simplifying fractions and dividing quantities into equal parts.
LCM(a, b) = (a × b) / HCF(a, b)
LCM is the smallest number that is a multiple of both a and b. It is essential for finding common denominators in fractions.
Prime Factorisation: n = p₁^a × p₂^b × ... × pₖ^c
Any integer n can be expressed as a product of prime factors raised to their respective powers, assisting in finding factors and multiples effectively.
Factor x = {f | f is a divisor of n}
The set of factors of a number n includes all numbers that divide n evenly. This concept aids in listing common factors for HCF.
Multiples of n = {n, 2n, 3n, ...}
Multiples of a number n are generated by multiplying n with whole numbers. This principle helps in finding the LCM.
Common Factors = {f | f divides both a and b}
This notation represents the set of factors that are shared between a and b. Identifying these is crucial for HCF calculations.
Greatest Common Divisor (GCD) = HCF
GCD and HCF are interchangeable terms referring to the largest factor common to two or more numbers.
If n is a multiple of m, then HCF(m, n) = m
This property shows that if one number is a multiple of another, then the smaller number is the HCF of both.
For prime numbers, HCF = 1
If two numbers share no common prime factors, their HCF is 1, indicating they are coprime.
For any number, factors = {d | d < n and d divides n}
This representation shows that factors of n are all divisors less than n, aiding in efficient factor listing.
Equations
12 ft = 4 ft × 3
This equation illustrates that the breadth of the room (12 ft) can be reached by using three tiles of 4 ft each. It demonstrates how to calculate the number of tiles required.
16 ft = 4 ft × 4
Similarly, the length of the room (16 ft) can be fully covered with four tiles of size 4 ft, emphasizing the efficiency of using the largest tile size.
Common factors of 84 and 108 = {1, 2, 3, 4, 6, 12}
This equation lists the common factors between the two numbers, necessary to determine the optimal bag weight for packing rice.
HCF(45, 75) = 15
Finding the HCF of these two numbers provides their highest common factor, essential in problems involving shared quantities.
4 is the HCF of 12 and 16
This equation states that the highest common factor for the room dimensions is 4, which guides the selection of tile size.
LCM(10, 7) is the first number both multiples share = 70
This equation finds the least common multiple of Kabamai's 10-day schedule and the sweet shop's 7-day cycle.
2 × 3 × 5 = 30, factors of 30
This shows the breakdown of 30 into its prime factors, aiding in determining all other factors related to it.
96 = 2^5 × 3^1
The prime factorization approach provides the breakdown of a number into its prime components, simplifying HCF and LCM calculations.
70 = (2 × 5 × 7)
This expression denotes the LCM of 14 and 35, signifying the lowest shared multiple relevant in scenarios of combined events.
Factors of 225 = {1, 3, 5, 9, 15, 25, 45, 75, 225}
This equation provides a complete list of factors for 225 through systematic prime factorization.
