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

If a right triangle has legs of 9 cm and 12 cm, what is the length of the hypotenuse?

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If the perimeter of a square is 36 cm, what is the length of its diagonal?

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The probability of getting an even number when rolling a die is:

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The sides of a triangle are 13 cm, 14 cm, and 15 cm. What is its area?

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If two complementary angles differ by 30°, what are the angles?

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The LCM of two numbers is 60, and their HCF is 5. If one of the numbers is 20, what is the other number?

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

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If the length of a rectangle is doubled and the width is halved, what is the change in area?

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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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