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If K is the least positive integer that is divisible by every integer
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09 Jan 2026, 09:57
09 Jan 2026, 09:57
Expert Reply
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Question Stats:
75% (01:21) correct
25% (02:05) wrong based on 4 sessions
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If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then $K=$
◯ 840
◯ 2,520
◯ 6,720
◯ 20,160
◯ 40,320
◯ 840
◯ 2,520
◯ 6,720
◯ 20,160
◯ 40,320
- Part of the project: GRE Daily Quant Challenge - Topic-Wise 2025!!!
- Also replying to the unanswered questions
Re: If K is the least positive integer that is divisible by every integer
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25 Jan 2026, 07:41
25 Jan 2026, 07:41
Expert Reply
To find the least positive integer $K$ that is divisible by every integer from 1 to 8 inclusive, we need to find the Least Common Multiple (LCM) of the set $\(\{1,2,3,4,5,6,7,8\}\)$.
1. Find the Prime Factorization of each number:
- $1=1$
- $\(2=2^1\)$
- $\(3=3^1\)$
- $\(4=2^2\)$
- $\(5=5^1\)$
- $\(6=2^1 \times 3^1\)$
- $\(7=7^1\)$
- $\(8=2^3\)$
2. Identify the highest power of each prime factor:
To find the LCM, we take each prime that appears in any of the factorizations and raise it to the highest power found:
- Prime 2: The highest power is $\(2^3=8\)$.
- Prime 3: The highest power is $\(3^1=3\)$.
- Prime 5: The highest power is $\(5^1=5\)$.
- Prime 7: The highest power is $\(7^1=7\)$.
3. Calculate the product:
$$
\(\begin{gathered}
K=8 \times 3 \times 5 \times 7 \\
K=24 \times 35 \\
K=840
\end{gathered}\)
$$
Conclusion: The value of $K$ is $\(\mathbf{8 4 0}\)$.
1. Find the Prime Factorization of each number:
- $1=1$
- $\(2=2^1\)$
- $\(3=3^1\)$
- $\(4=2^2\)$
- $\(5=5^1\)$
- $\(6=2^1 \times 3^1\)$
- $\(7=7^1\)$
- $\(8=2^3\)$
2. Identify the highest power of each prime factor:
To find the LCM, we take each prime that appears in any of the factorizations and raise it to the highest power found:
- Prime 2: The highest power is $\(2^3=8\)$.
- Prime 3: The highest power is $\(3^1=3\)$.
- Prime 5: The highest power is $\(5^1=5\)$.
- Prime 7: The highest power is $\(7^1=7\)$.
3. Calculate the product:
$$
\(\begin{gathered}
K=8 \times 3 \times 5 \times 7 \\
K=24 \times 35 \\
K=840
\end{gathered}\)
$$
Conclusion: The value of $K$ is $\(\mathbf{8 4 0}\)$.
gmatclubot
Re: If K is the least positive integer that is divisible by every integer [#permalink]
25 Jan 2026, 07:41
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