Topic Details (Notes format)

Introduction to Euler’s Totient Function (φ)

Subject: Mathematics

Book: Maths Mastery

Euler’s Totient Function φ(n) counts how many integers ≤n are coprime to n. For prime p, φ(p)=p–1. For example, φ(8)=4 because only {1,3,5,7} are coprime with 8. This function is core in number theory and cryptography (Euler’s theorem, RSA encryption). Euler’s theorem states a^φ(n)≡1 (mod n) if gcd(a,n)=1. Understanding φ fosters advanced integer analysis, letting you compute exponents mod n or analyze prime-based structures. Mastery in totient calculations links to deeper insights in modern computer security and theoretical math.

Practice Questions

A cone has a base radius of 7 cm and height of 24 cm. What is its volume?

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The sum of the reciprocals of two numbers is 1/4. If one number is 12, what is the other?

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The perimeter of a rectangle is 40 cm, and its length is 12 cm. What is its width?

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What is the area of an equilateral triangle with side length 10 cm?

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If a cone has a radius of 5 cm and a height of 12 cm, what is its slant height?

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If a number is divisible by 9, it is also divisible by which of the following?

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If the perimeter of a square is 40 cm, what is the area of the square?

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A sum of money triples itself in 12 years at simple interest. What is the rate of interest per annum?

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What is the greatest common divisor (GCD) of 36 and 48?

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If a:b = 7:9 and b:c = 5:6, what is a:c?

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