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Zorluk: ZorFactors, Multiples, and Prime Factorization

A logistics company assigns identification codes to container shipments based on a positive integer NN. The number NN has exactly 2424 positive factors, and its prime factorization contains only the prime factors 22, 33, and 77. If NN is a multiple of 1414 and the number of odd positive factors of NN is 66, what is the smallest possible value of NN?

  1. 504504Cevap
  2. B
    11761176
  3. C
    252252
  4. D
    15121512

Cevap

The smallest possible value of NN is 504504.
The number 504504 has prime factorization 23×32×712^3 \times 3^2 \times 7^1. Total positive factors = (3+1)(2+1)(1+1)=24(3+1)(2+1)(1+1) = 24, and odd positive factors = (2+1)(1+1)=6(2+1)(1+1) = 6. It is divisible by 1414 (504=14×36504 = 14 \times 36) and is the smaller of the two valid numbers (504504 and 11761176).

Adım Adım Çözüm

1
Express NN in terms of its prime factors.
N=2a×3b×7cN = 2^a \times 3^b \times 7^c, where a≥1,b≥1,c≥1a \ge 1, b \ge 1, c \ge 1 as NN contains prime factors 2,3,2, 3, and 77, and is a multiple of 14=2×714 = 2 \times 7.
The question specifies that NN is divisible by 1414 and contains prime factors 22, 33, and 77.
2
Formulate equations for total factors and odd factors.
Total positive factors T(N)=(a+1)(b+1)(c+1)=24T(N) = (a + 1)(b + 1)(c + 1) = 24. Odd positive factors depend only on odd prime powers (3b×7c3^b \times 7^c), so (b+1)(c+1)=6(b + 1)(c + 1) = 6.
Odd factors correspond to choosing 20=12^0 = 1 for the even prime factor.
3
Solve for exponent aa.
(a+1)×6=24  ⟹  a+1=4  ⟹  a=3(a + 1) \times 6 = 24 \implies a + 1 = 4 \implies a = 3.
Dividing total factors by odd factors isolates the term (a+1)(a + 1).
4
Find possible values for bb and cc.
Since (b+1)(c+1)=6(b + 1)(c + 1) = 6 with b,c≥1b, c \ge 1, the possible factor pairs for (b+1,c+1)(b+1, c+1) are (2,3)(2, 3) or (3,2)(3, 2). Case 1: b=1,c=2  ⟹  N=23×31×72=1176b=1, c=2 \implies N = 2^3 \times 3^1 \times 7^2 = 1176. Case 2: b=2,c=1  ⟹  N=23×32×71=504b=2, c=1 \implies N = 2^3 \times 3^2 \times 7^1 = 504.
Integer factorizations of 66 give two distinct exponent assignments.
5
Determine the minimum value.
Comparing 504504 and 11761176, the minimum value is 504504.
Assigning the larger exponent to the smaller prime base (33 instead of 77) minimizes the product.

Anahtar Kavram

Factors and Prime Factorization Constraints
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Factors, Multiples, and Prime Factorization

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