Multiple choice

Let x be the least number divisible by 16, 24, 30, 36 and 45, and x is also a perfect square. What is the remainder when x is divided by 123?

  1. 100

  2. 40

  3. 103

  4. 33
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D Correct answer
Explanation

LCM of 16, 24, 30, 36, 45: 16=2^4, 24=2^3*3, 30=2*3*5, 36=2^2*3^2, 45=3^2*5. LCM = 2^4 * 3^2 * 5 = 16 * 9 * 5 = 720. To be a perfect square, multiply by 5: 720 * 5 = 3600. 3600 / 123: 3600 = 123 * 29 + 33. Remainder is 33.

AI explanation

To find the least common multiple of 16, 24, 30, 36, and 45 using prime factorization, we take the highest power of each prime present, giving 2^4 times 3^2 times 5^1 = 720. For this least common multiple to be a perfect square, all prime factors must have even exponents, so we multiply by 5 to get x = 3600. Dividing 3600 by 123 gives a quotient of 29 and a remainder of 33.

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