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' $K$ ' is the least positive integer that is divisible by all positiv
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05 Apr 2025, 01:03
05 Apr 2025, 01:03
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Question Stats:
37% (01:03) correct
62% (02:23) wrong based on 8 sessions
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' $K$ ' is the least positive integer that is divisible by all positive integers less than or equal to 10 . Find the total number of factors of $k$ (excluding 1 and ' $k$ ').
A. 4
B. 6
C. 7
D. 46
E. 48
A. 4
B. 6
C. 7
D. 46
E. 48
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Re: ' $K$ ' is the least positive integer that is divisible by all positiv
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09 Apr 2025, 04:00
09 Apr 2025, 04:00
Expert Reply
The least positive integer divisible by all the positive integers less than or equal to 10 is the $\(\operatorname{LCM}\)$ of $\(\{1,2,3,4,5,6,7$, $8,9\)$, and \(10$\}\)\(=2520=2^3 \times 3^2 \times 5 \times 7=K\)$.
Therefore, the total number of factors of $K$ (including 1 and $K)$ is $\((3+1)(2+1)(1+1)(1+1)=4 \times 3 \times 2 \times 2=48\)$.
Therefore, if we exclude 1 and $K$, the total number of factors turns out to be $\(48-2=46\)$.
Thus, the correct answer choice is $D$.
Therefore, the total number of factors of $K$ (including 1 and $K)$ is $\((3+1)(2+1)(1+1)(1+1)=4 \times 3 \times 2 \times 2=48\)$.
Therefore, if we exclude 1 and $K$, the total number of factors turns out to be $\(48-2=46\)$.
Thus, the correct answer choice is $D$.
gmatclubot
Re: ' $K$ ' is the least positive integer that is divisible by all positiv [#permalink]
09 Apr 2025, 04:00
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