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Zorluk: OrtaPrime Factorization, GCD, and LCM

What is the smallest positive integer that is a multiple of 18, 24, and 30, and is also a perfect square?

Cevap: 3600

Cevap

The smallest positive integer that is a multiple of 18, 24, and 30, and is also a perfect square is 3600.
The least common multiple of 18, 24, and 30 is 360, which factors into 23⋅32⋅512^3 \cdot 3^2 \cdot 5^1. For an integer to be a perfect square, all exponents in its prime factorization must be even. Multiplying 360 by 2⋅5=102 \cdot 5 = 10 completes the odd exponents to even values (24⋅32⋅522^4 \cdot 3^2 \cdot 5^2), giving 3600, which is the smallest perfect square divisible by 18, 24, and 30.

Adım Adım Çözüm

1
Find the prime factorization of 18, 24, and 30
18=21⋅3218 = 2^1 \cdot 3^2, 24=23⋅3124 = 2^3 \cdot 3^1, 30=21⋅31⋅5130 = 2^1 \cdot 3^1 \cdot 5^1
Decomposing numbers into prime factors allows calculation of the LCM and analysis of perfect square conditions.
2
Calculate the LCM of 18, 24, and 30
LCM(18,24,30)=23⋅32⋅51=360\text{LCM}(18, 24, 30) = 2^3 \cdot 3^2 \cdot 5^1 = 360
Any common multiple must be a multiple of the LCM of these three numbers.
3
Determine the smallest factor required to make the prime exponents even
Multiply 360 by 21⋅51=102^1 \cdot 5^1 = 10
A perfect square requires all prime exponents to be even; 2 has power 3 and 5 has power 1 in 360, so one more factor of 2 and one more factor of 5 are required.
4
Compute the final result
360×10=3600360 \times 10 = 3600
3600=24⋅32⋅52=6023600 = 2^4 \cdot 3^2 \cdot 5^2 = 60^2, which is a perfect square.

Anahtar Kavram

Prime Factorization, LCM, and Exponent Properties of Perfect Squares
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Prime Factorization, GCD, and LCM

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