The LCM of 48, 72, and another number, X, is 576. Which of the values given below can be the value of X?

Explanation

To find the possible value of XXX given that the least common multiple (LCM) of 484848, 727272, and XXX is 576576576, let's go through the steps systematically. Step 1: Prime Factorization First, let’s factorize each number involved: 1. 48: 48=24×348 = 2^4 \times 348=24×3 2. 72: 72=23×3272 = 2^3 \times 3^272=23×32 3. 576: 576=26×32576 = 2^6 \times 3^2576=26×32 Step 2: Analyze the LCM Requirement The LCM of 484848, 727272, and XXX must be 576=26×32576 = 2^6 \times 3^2576=26×32. For the LCM to equal 576576576: The highest power of 222 among 484848, 727272, and XXX must be 262^626. The highest power of 333 among 484848, 727272, and XXX must be 323^232. Step 3: Determine Possible Values for XXX 1. Since 484848 provides 242^424 and 727272 provides 232^323, XXX must provide at least 262^626 to meet the LCM requirement for the factor of 222. 2. Similarly, for the factor of 333, XXX should not exceed 323^232, as 323^232 is already covered by 727272. Step 4: Find the Smallest Possible XXX that Meets These Requirements To satisfy both conditions: XXX must have a factor of 262^626, since neither 484848 nor 727272 alone has this power. XXX can include 303^030, 313^131, or 323^232, as adding more would exceed the required 323^232 for the LCM. This gives possible values of XXX as: 1. 26=642^6 = 6426=64 2. 26×3=1922^6 \times 3 = 19226×3=192 3. 26×32=5762^6 \times 3^2 = 57626×32=576 Conclusion The possible values for XXX that satisfy the condition of an LCM of 576576576 are: 64 192 576 Among these, the answer will depend on the specific options provided.