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Zorluk: OrtaFactors, Multiples, and Prime Factorization

A digital encryption protocol derives a validation key from the mathematical properties of the integer 1260012600. The key is calculated by multiplying the total quantity of its even positive divisors by the total quantity of its odd positive divisors. What is the exact value of this validation key?

Cevap: 972

Cevap

972
By finding the prime factorization of 1260012600 as 23×32×52×712^3 \times 3^2 \times 5^2 \times 7^1, we can separate the divisors into even and odd categories. Odd divisors utilize only the odd prime factors, yielding a total of (2+1)(2+1)(1+1)=18(2+1)(2+1)(1+1) = 18 odd divisors. Even divisors require at least one power of 22, giving 3×(2+1)(2+1)(1+1)=543 \times (2+1)(2+1)(1+1) = 54 even divisors. Following the protocol's rule, their product is 18×54=97218 \times 54 = 972.

Adım Adım Çözüm

1
Find the prime factorization of 1260012600.
12600=23×32×52×7112600 = 2^3 \times 3^2 \times 5^2 \times 7^1
Prime factorization is necessary to determine the number and types of divisors an integer has.
2
Calculate the quantity of odd positive divisors.
Number of odd divisors = 1818
Odd divisors are formed by using only the odd prime factors (33, 55, and 77). By adding 11 to each of their exponents and multiplying them, we get (2+1)×(2+1)×(1+1)=3×3×2=18(2+1) \times (2+1) \times (1+1) = 3 \times 3 \times 2 = 18.
3
Calculate the quantity of even positive divisors.
Number of even divisors = 5454
Even divisors must include at least one factor of 22. The number of choices for the power of 22 is equal to its exponent (33 choices: 21,22,232^1, 2^2, 2^3). Multiplying this by the choices for odd prime factors gives 3×(2+1)×(2+1)×(1+1)=3×3×3×2=543 \times (2+1) \times (2+1) \times (1+1) = 3 \times 3 \times 3 \times 2 = 54.
4
Multiply the number of even divisors by the number of odd divisors to find the validation key.
54×18=97254 \times 18 = 972
The encryption protocol specifically defines the key as the product of these two calculated quantities.

Anahtar Kavram

Prime Factorization and Number of Divisors (Even and Odd)
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Konular

Factors, Multiples, and Prime Factorization

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