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Zorluk: OrtaFactors, Multiples, and Prime Factorization

Consider the positive integer N=3600N = 3600. What is the total number of positive divisors of NN that are multiples of either 44 or 99?

  1. A
    12
  2. B
    42
  3. 33Cevap
  4. D
    9

Cevap

The total number of positive divisors of 3600 that are multiples of either 4 or 9 is 33.
The correct answer is derived by finding the prime factorization of 3600 (24×32×522^4 \times 3^2 \times 5^2), determining the separate counts of divisors that are multiples of 4 (27 factors) and multiples of 9 (15 factors), and then using the inclusion-exclusion principle to subtract the intersection (divisors that are multiples of 36, which is 9 factors). This yields 27+15−9=3327 + 15 - 9 = 33.

Adım Adım Çözüm

1
Prime factorize 3600 to identify the structure of its divisors.
3600=36×100=24×32×523600 = 36 \times 100 = 2^4 \times 3^2 \times 5^2.
A divisor's properties depend entirely on the combinations of its prime factors.
2
Calculate the total number of divisors that are multiples of 4.
Multiples of 4 must contain at least 222^2. The valid powers for 2 are {22,23,24}\{2^2, 2^3, 2^4\} (3 choices). For 3, {30,31,32}\{3^0, 3^1, 3^2\} (3 choices). For 5, {50,51,52}\{5^0, 5^1, 5^2\} (3 choices). Total = 3×3×3=273 \times 3 \times 3 = 27 divisors.
To be a multiple of 4, the divisor's prime factorization must contain a power of 2 that is at least 2.
3
Calculate the total number of divisors that are multiples of 9.
Multiples of 9 must contain at least 323^2. The valid powers for 2 are 5 choices (from 0 to 4). For 3, {32}\{3^2\} (1 choice). For 5, 3 choices. Total = 5×1×3=155 \times 1 \times 3 = 15 divisors.
To be a multiple of 9, the divisor must contain a power of 3 that is at least 2.
4
Calculate the number of divisors that are multiples of both 4 and 9 (i.e., multiples of 36).
These divisors must contain at least 222^2 and 323^2. Choices: 3 (for 2) ×\times 1 (for 3) ×\times 3 (for 5) = 99 divisors.
Since 4 and 9 are co-prime, multiples of both must be multiples of their Least Common Multiple (LCM), which is 36.
5
Apply the Principle of Inclusion-Exclusion to find divisors that are multiples of either 4 or 9.
27+15−9=3327 + 15 - 9 = 33 divisors.
Adding the multiples of 4 and 9 counts their intersection (multiples of 36) twice, so it must be subtracted once to find the correct union.

Anahtar Kavram

Applying set theory (inclusion-exclusion principle) to the combinations of prime factors of a number to find constrained subsets of divisors.
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Konular

Factors, Multiples, and Prime Factorization

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