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Answer :
Final answer:
The smallest possible value of x for which the lowest common multiple of x and 180 is 360, is 120. This is because 120 includes the necessary additional factor of 2, which, in combination with the factors of 180, adds up to the LCM of 360.
Explanation:
The student has asked to find the smallest possible value of x such that the lowest common multiple (LCM) of x and 180 is 360. To find the LCM, each number must be broken down into its prime factors. The number 180 can be factored into 2² × 3² × 5. For the LCM of x and 180 to be 360, the prime factors of x must include 2³ since 360 includes an additional factor of 2 compared to 180. Therefore, the smallest possible value for x must share all the prime factors of 180 and also include the factor 2 once more to account for 2³ in 360.
The options given are:
90, which factors as 2¹ × 3² × 5.
120, which factors as 2³ × 3± × 5.
180, which factors as 2² × 3² × 5.
360, which factors as 2³ × 3² × 5.
By comparing the options, we find that option b. 120 is the smallest number with 2³ as a prime factor and is the smallest value that when combined with 180 gives 360 as the LCM.
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Final answer:
The smallest possible value of x for which the LCM of x and 180 is 360 is 120. Option b is the correct answer because 120 is not a factor of 180 and the LCM of 120 and 180 is 360.
Explanation:
Finding the Smallest Possible Value of x
To solve for the smallest possible value of x given that the lowest common multiple (LCM) of x and 180 is 360, we need to find a value for x that is a factor of 360. It also needs to be true that there are no smaller multiples of x that are also multiples of 180. We see that 180 is already a factor of 360, so the LCM of 180 and itself is 180, not 360, meaning c. 180 is not the answer we seek.
The option a. 90 is half of 180 and a factor of 360, which means that the LCM of 90 and 180 is also 180, not 360. So option a is not correct either.
Now considering option b. 120, since 120 is not a factor of 180 and the LCM of 120 and 180 is indeed 360, this is the smallest possible value of x that satisfies the condition. Therefore, the smallest possible value of x is 120.
