LCM addiction

Number Theory Level 3

If
A=997! and
B=1001 *1002 *1003 *1004 *1005 *1006.
Lcm of A and B is k!.
Find the value of k?


The answer is 997.

Abhishek Singh by Abhishek Singh , 24, India

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2 solutions

Kay Xspre
Nov 15, 2015

B can be written as the following.

b = ( 143 × 7 ) × ( 167 × 6 ) × ( 59 × 17 ) × ( 251 × 4 ) × ( 67 × 15 ) × ( 2 × 503 ) b = (143\times7)\times(167\times6)\times(59\times17)\times(251\times 4)\times(67\times15)\times(2\times503)

Or simply rearranging to b = 2 × 4 × 6 × 7 × 15 × 17 × 59 × 67 × 143 × 167 × 251 × 503 b = 2\times4\times6\times7\times15\times17\times59\times67\times143\times167\times251\times503

As we realized that a = 997 ! = 1 × 2 × 3 × 4 × ⋯ × 997 a = 997! = 1\times2\times3\times4\times\dots\times997 , a a will contain all factors in b b , and thus, a a is the LCM for A and B.

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Abhishek Singh
Nov 14, 2015

Since none of the 6 numbers in B is prime....So,the answer of LCM is 997! ..But this method doesn't work in the case of repeating factors as contradicted by Mr. Harish Sasikumar.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

That cannot be the complete solution. Let's say when A=5! and B= 10 × 9 × 8 10\times9\times8 , LCM is not 5! eventhough none of the 3 numbers in B is a prime.

Harish Sasikumar - 5 years, 7 months ago

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