Question

What is the quotient if the least common multiple of the first 40 positive integers divided by the least common multiple of the first 30 positive integers.
Give detailed explanation
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Answer 1

Answer:

LCM of first 40 positive integers is

5342931457063200

LCM of first 30 positive integers is

2329089562800

The required guantity is

5342931457063200÷2329089562800

Which is equal to

2294 answer

Answer 2

Answer:

LCM(1,2,3,....40) / LCM( 1 to 30)=2294

Step-by-step explanation:

From 1 to 30:

Max power of prime numbers

2=>16=2^4

3=>27=3³

5=> 25=5²

7=7^1

11=11

13 =>13

17 => 17

19=>19

23=>23

29=>29

Thus y=LCM (1,2,3......30)

y=2^4 * 3³*5²*7*11*13*17*19*23*29

Similarly x=LCM(1,2,3,..............40)

For finding prime factors we add in x the prime numbers from 31 to 40

So x=y*31*37*2 ( in 1to 30, max power iof 2 is 2^4=16  but in 1 to 40 2^5=32 is available so we have to multiply by 2^5/2^4=2

Thus x=LCM( 1,2,3.......40)=2^5 * 3³*5²*7*11*13*17*19*23*29*31*37

Thus x/y=y*2*31*37 /y

=2*31*37

=2294