Euler's Totient Function Calculator

Enter a number to calculate its Euler's Totient Function value.

Calculate the Value of Euler's Totient Function

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What Is Euler's Totient Function?

Euler's Totient Function, denoted as \( \phi(n) \), is a fundamental concept in number theory. It calculates the count of positive integers less than or equal to \( n \) that are coprime with \( n \) (i.e., integers that share no common factors with \( n \) other than 1).

How to Calculate Euler's Totient Function?

Factorize \( n \): Decompose \( n \) into its prime factors: \( n = p_1^{k_1} \times p_2^{k_2} \times \ldots \times p_m^{k_m} \) Here, \( p_1, p_2, \ldots, p_m \) are the distinct prime factors of \( n \).
Apply the Formula: The Euler's Totient Function is calculated as: \( \phi(n) = n \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right) \ldots \left(1 - \frac{1}{p_m}\right) \)

Examples

Example 1: Calculate \( \phi(12) \)

Solution:

Prime factorization: \( 12 = 2^2 \times 3^1 \)

Applying the formula:

\( \phi(12) = 12 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 12 \times \frac{1}{2} \times \frac{2}{3} = 4 \)

Result: \( \phi(12) = 4 \)

Example 2: Calculate \( \phi(30) \)

Solution:

Prime factorization: \( 30 = 2^1 \times 3^1 \times 5^1 \)

Applying the formula:

\( \phi(30) = 30 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{5}\right) = 30 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} = 8 \)

Result: \( \phi(30) = 8 \)

Example 3: Calculate \( \phi(360) \)

Solution:

Prime factorization: \( 360 = 2^3 \times 3^2 \times 5^1 \)

Applying the formula:

\( \phi(360) = 360 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{5}\right) = 360 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} = 96 \)

Result: \( \phi(360) = 96 \)

First 100 Euler's Totient Function Values

φ(1) = 1
φ(2) = 1
φ(3) = 2
φ(4) = 2
φ(5) = 4
φ(6) = 2
φ(7) = 6
φ(8) = 4
φ(9) = 6
φ(10) = 4
φ(11) = 10
φ(12) = 4
φ(13) = 12
φ(14) = 6
φ(15) = 8
φ(16) = 8
φ(17) = 16
φ(18) = 6
φ(19) = 18
φ(20) = 8

φ(21) = 12
φ(22) = 10
φ(23) = 22
φ(24) = 8
φ(25) = 20
φ(26) = 12
φ(27) = 18
φ(28) = 12
φ(29) = 28
φ(30) = 8
φ(31) = 30
φ(32) = 16
φ(33) = 20
φ(34) = 16
φ(35) = 24
φ(36) = 12
φ(37) = 36
φ(38) = 18
φ(39) = 24
φ(40) = 16

φ(41) = 40
φ(42) = 12
φ(43) = 42
φ(44) = 20
φ(45) = 24
φ(46) = 22
φ(47) = 46
φ(48) = 16
φ(49) = 42
φ(50) = 20
φ(51) = 32
φ(52) = 24
φ(53) = 52
φ(54) = 18
φ(55) = 40
φ(56) = 24
φ(57) = 36
φ(58) = 28
φ(59) = 58
φ(60) = 16

φ(61) = 60
φ(62) = 30
φ(63) = 36
φ(64) = 32
φ(65) = 48
φ(66) = 20
φ(67) = 66
φ(68) = 32
φ(69) = 44
φ(70) = 24
φ(71) = 70
φ(72) = 24
φ(73) = 72
φ(74) = 36
φ(75) = 40
φ(76) = 36
φ(77) = 60
φ(78) = 24
φ(79) = 78
φ(80) = 32

φ(81) = 54
φ(82) = 40
φ(83) = 82
φ(84) = 24
φ(85) = 64
φ(86) = 42
φ(87) = 56
φ(88) = 40
φ(89) = 88
φ(90) = 24
φ(91) = 72
φ(92) = 44
φ(93) = 60
φ(94) = 46
φ(95) = 72
φ(96) = 32
φ(97) = 96
φ(98) = 42
φ(99) = 60
φ(100) = 40

Euler's Totient Function Calculator

Calculate the Value of Euler's Totient Function

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What Is Euler's Totient Function?

How to Calculate Euler's Totient Function?

Examples

Example 1: Calculate \( \phi(12) \)

Example 2: Calculate \( \phi(30) \)

Example 3: Calculate \( \phi(360) \)

First 100 Euler's Totient Function Values

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