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Sample Question 8:
Step 1: We start by defining the Least Common Multiple (LCM). The LCM of a set of numbers is obtained by taking the greatest powers of the prime numbers in the prime factorization of all the numbers.
Step 2: Applying this rule, we determine that \(M = 2^4 \cdot 3^3 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29.\), which is the least common multiple of all the integers from 10 to 30 inclusive.
Step 3: We then find the value of \(N\), which is the LCM of \(M\), 32, 33, 34, 35, 36, 37, 38, 39, and 40. Following the same rule, \(N = M \cdot 2 \cdot 37\). This is because there is an additional power of \(2\) and an additional power of \(37\) in the prime factorization of the set of numbers.
Step 4: Finally, we compute the value of \(\frac{N}{M}\) by substitute the found expression for \(N\) and \(M\). This becomes \(\frac{N}{M} = 2\cdot 37 = 74\).
So, the answer is \(\textbf{(D)}\ 74\).
Let \(M\) be the least common multiple of all the integers \(10\) through \(30,\) inclusive. Let \(N\) be the least common multiple of \(M,32,33,34,35,36,37,38,39,\) and \(40.\) What is the value of \(\frac{N}{M}?\)
\(\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 37 \qquad\textbf{(D)}\ 74 \qquad\textbf{(E)}\ 2886\)
Answer Keys
Question 8: DSolutions
Question 8Step 1: We start by defining the Least Common Multiple (LCM). The LCM of a set of numbers is obtained by taking the greatest powers of the prime numbers in the prime factorization of all the numbers.
Step 2: Applying this rule, we determine that \(M = 2^4 \cdot 3^3 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29.\), which is the least common multiple of all the integers from 10 to 30 inclusive.
Step 3: We then find the value of \(N\), which is the LCM of \(M\), 32, 33, 34, 35, 36, 37, 38, 39, and 40. Following the same rule, \(N = M \cdot 2 \cdot 37\). This is because there is an additional power of \(2\) and an additional power of \(37\) in the prime factorization of the set of numbers.
Step 4: Finally, we compute the value of \(\frac{N}{M}\) by substitute the found expression for \(N\) and \(M\). This becomes \(\frac{N}{M} = 2\cdot 37 = 74\).
So, the answer is \(\textbf{(D)}\ 74\).