If a number be divided into its prime factors, or reduced to a product of prime numbers only (as in 360 = 2 × 2 × 2 × 3 × 3 × 5), and if _a_, _b_, _c_, &c. be the prime factors, and α, β, γ, &c. the number of times they severally enter, so that the number is _a_{^α} × _b_ᵝ × _c_ᵞ × &c., then this can be done in only one way: For any prime number _v_, not included in the above list, is prime to _a_, and therefore to _a_{^α}, to _b_ and therefore to _b_ᵝ and therefore to _a_{^α} × _b_ᵝ Proceeding in this way, we prove that _v_ is prime to the complete product above, or to the given number itself. The number of divisors which the preceding number _a_{^α}_b_ᵝ_c_ᵞ ... can have, 0 and itself included, is (α + 1)(β+ 1)(γ + 1).... For _a_{^α} as the divisors 1, _a_, _a_² ... _a_{^α} and no others, α + 1 in all. Similarly, _b_ᵝ has β+ 1 divisors, and so on. Now as all the divisors are made by multiplying together one out of each set, their number (page 202) is (α + 1)(β + 1)(γ+ 1).... If a number, _n_, be divisible by certain prime numbers, say 3, 5, 7, 11, then the third part of all the numbers up to _n_ is divisible by 3, the fifth part by 5, and so on. But more than this: when the multiples of 3 are omitted, exactly the fifth part of _those which remain_ are divisible by 5; for the fifth part of the whole are divisible by 5, and the fifth part of those which are removed are divisible by 5, therefore the fifth part of those which are left are divisible by 5. Again, because the seventh part of the whole are divisible by 7, and the seventh part of those which are divisible by 3, or by 5, or by 15, it follows that when all those which are multiples of 3 or 5, or both, are removed, the seventh part of those which remain are divisible by 7; and so on. Hence, the number of numbers not exceeding n, which are not divisible by 3, 5, 7, or 11, is ¹⁰/₁₁ of ⁶/₇ of ⁴/₅ of ²/₃ of n. Proceeding in this way, we find that the number of numbers which are prime to _n_, that is, which are not divisible by any one of its prime factors, _a_, _b_, _c_, ... is _a_ - 1 _b_ - 1 _c_ - 1 _n_ ------- ------- ------- ... _a_ _b_ _c_ or _a_{^α-1} - 1}_b_ᵝ⁻¹_c_ᵞ⁻¹ ... (_a_ - 1)(_b_ - 1)(_c_ - 1).... Thus, 360 being 2³3²5, its number of divisors is 4 × 3 × 2, or 24, and there are 2³3.1.2.4 or 96 numbers less than 360 which are prime to it. PROP. 5. If _a_ be prime to _b_, then the terms of the series, _a_, _a_², _a_³, ... severally divided by _b_, must all leave different remainders, until 1 occurs as a remainder, after which the cycle of remainders will be again repeated.