What is financial infrastructure?

The network of banks, payment systems and stock markets that helps manage money

What are the main things a bank does?

Holds deposits and lends money as loans

How do banks make money?

They pay lower interest on deposits and charge higher interest on loans

Earning interest on previous interest as well as on the original amount

What is the HCF of two numbers?

The greatest factor common to both numbers.

What is the LCM of two numbers?

The smallest multiple common to both numbers.

Find the HCF of 30 and 72.

Taking the minimum shared primes, HCF \(= 2 \times 3 = 6\).

Find the LCM of 14 and 35.

Taking the maximum occurrences, LCM \(= 2 \times 5 \times 7 = 70\).

If one number is a factor of another, what is their HCF?

The smaller number is their HCF (and the larger is their LCM).

What is a conjecture in mathematics?

A claim made without proof, which may be disproved by a counterexample.

Finding Common Ground

Q: Write the prime factorisation of 90.

\(90 = 2 \times 3 \times 3 \times 5\).

Q: What is the relation between HCF, LCM and the product of two numbers?

\(\text{HCF} \times \text{LCM}\) equals the product of the two numbers.

Class 7 Mathematics · Ganita Prakash Part 2 · Chapter 3

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CBSE Class 7 · Mathematics

8 chapters: lakhs and crores to fractions

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Triangle inequality, Ch 7
Angle pairs of parallel lines, Ch 5
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Magic squares and Virahāṅka numbers, Ch 6
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Five congruence conditions to equations

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HCF and LCM by prime factors, Ch 3
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Mean, median and outliers, Ch 5
Ruler-and-compass constructions, Ch 6

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Summary

This chapter is about the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM), built on prime factorisation. A prime has only 1 and itself as factors; every number can be written as a product of primes in essentially one way (e.g. \(90 = 2\times3\times3\times5\)), found efficiently by the division method. From a prime factorisation you can list all factors as products of "subparts". The HCF is the largest common factor — taken by including the minimum number of times each shared prime occurs across the numbers — while the LCM is the smallest common multiple — taken by including the maximum number of occurrences of each prime. You learn a quick column method that produces both at once, and explore general patterns (the HCF of two consecutive numbers is 1; if one number is a factor of another, it is their HCF and the larger is their LCM). A beautiful relation ties them together: for any two numbers, \(\text{HCF}\times\text{LCM}\) equals the product of the two numbers. The chapter also introduces the ideas of conjecture, counterexample and generalisation.

Primes and prime factorisationListing factors from prime factorsFinding the HCFFinding the LCMHCF–LCM relation, patterns and generalisation

Key terms

Prime number

A number greater than 1 whose only factors are 1 and itself.

Prime factorisation

Writing a number as a product of primes, e.g. \(90 = 2\times3\times3\times5\).

Highest Common Factor (HCF)

The greatest factor common to two or more numbers; also the GCD.

Lowest Common Multiple (LCM)

The smallest multiple common to two or more numbers.

Co-prime numbers

Two numbers whose only common factor is 1, so their HCF is 1.

Conjecture

A statement believed true but not yet proved; disproved by a counterexample.

Extra questions & answers

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The HCF of two numbers is:

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