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Zorluk: OrtaFactors, Multiples, and Prime Factorization

If the least common multiple of a positive integer nn and 15 is 90, how many possible values are there for nn?

  1. A
    1
  2. 2Cevap
  3. C
    3
  4. D
    4
  5. E
    6

Cevap

The correct answer is 2.
The correct answer is 2. The prime factorization of 15 is 31×513^1 \times 5^1, and the prime factorization of 90 is 21×32×512^1 \times 3^2 \times 5^1. The least common multiple (LCM) of two numbers is found by taking the highest power of each prime factor present in their factorizations. Since 90 contains 212^1, and 15 does not, nn must contain exactly 212^1. Since 90 contains 323^2, and 15 only contains 313^1, nn must contain exactly 323^2. Since both 90 and 15 contain 515^1, the power of 5 in nn can be either 0 or 1. This results in two possible values: 21×32×50=182^1 \times 3^2 \times 5^0 = 18 and 21×32×51=902^1 \times 3^2 \times 5^1 = 90.

Adım Adım Çözüm

1
Write out the prime factorizations of 15 and 90.
15=31×5115 = 3^1 \times 5^1 and 90=21×32×5190 = 2^1 \times 3^2 \times 5^1.
To analyze the relationship between nn, 15, and their least common multiple (LCM), we must look at their prime factor components.
2
Determine the constraints on the exponents of the prime factors of nn using the definition of LCM.
The prime factorization of nn must be of the form 2a×3b×5c2^a \times 3^b \times 5^c. For the LCM of nn and 15 to be 90, the exponent of each prime factor in the LCM is the maximum of its exponents in nn and 15. Thus, we must have a=1a = 1, b=2b = 2, and c∈{0,1}c \in \{0, 1\}.
The LCM of two numbers takes the highest power of each prime factor present in both numbers.
3
Count the number of possible values for nn.
Since aa has 1 choice, bb has 1 choice, and cc has 2 choices, there are 1×1×2=21 \times 1 \times 2 = 2 possible values for nn (n=21×32×50=18n = 2^1 \times 3^2 \times 5^0 = 18 and n=21×32×51=90n = 2^1 \times 3^2 \times 5^1 = 90).
This gives the total number of distinct integers satisfying the condition.

Anahtar Kavram

The least common multiple of two numbers is found by taking the highest power of each prime factor present in the prime factorizations of both numbers.

Alternatif Yöntem

Alternatively, you can test the positive factors of 90 that are not factors of 15. The positive factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. By computing the least common multiple of each factor with 15, we find that only LCM(18, 15) = 90 and LCM(90, 15) = 90 satisfy the condition.
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Factors, Multiples, and Prime Factorization

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