Entre los siguientes números, encuentre los que son múltiplos de 2, 3, 5 y 10: 235, 360, 487, 564, 681, 792, 854, 904
Understand the Problem
La pregunta pide identificar, de una lista dada de números, aquellos que son múltiplos de 2, 3, 5 y 10 al mismo tiempo. Para que un número sea múltiplo de todos estos, debe ser múltiplo de su mínimo común múltiplo (mcm).
Answer
360
Answer for screen readers
360
Steps to Solve
- Find the Least Common Multiple (LCM) of 2, 3, 5, and 10
To be a multiple of 2, 3, 5, and 10, a number must be a multiple of their least common multiple (LCM). First, find the prime factorization of each number:
- 2 = 2
- 3 = 3
- 5 = 5
- 10 = 2 x 5 The LCM is found by taking the highest power of each prime factor present in the numbers: $$ LCM(2, 3, 5, 10) = 2 \times 3 \times 5 = 30 $$ So, we need to find numbers in the list that are multiples of 30.
- Check each number for divisibility by 30
Divide each number in the list by 30 and check if the result is an integer:
- 235 / 30 ≈ 7.83 (Not an integer)
- 360 / 30 = 12 (Integer)
- 487 / 30 ≈ 16.23 (Not an integer)
- 564 / 30 = 18.8 (Not an integer)
- 681 / 30 = 22.7 (Not an integer)
- 792 / 30 = 26.4 (Not an integer)
- 854 / 30 ≈ 28.47 (Not an integer)
- 904 / 30 ≈ 30.13 (Not an integer)
- Identify the Multiples of 30
From the above calculations, only 360 is divisible by 30.
360
More Information
The LCM of 2, 3, 5 and 10 is 30. Therefore, any multiple of 30 will be a multiple of all four numbers.
Tips
- Failing to calculate the LCM correctly. The LCM is crucial to simplifying the problem.
- Incorrectly performing the division to check for multiples.
- Missing that a multiple of 10 must end in a 0, which makes many of the numbers immediately ineligible
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