SOLUTION: determine the smallest 6 digit number which is exactly divisible by 8,15,and21

Question 499201: determine the smallest 6 digit number which is exactly divisible by 8,15,and21

Answer by oberobic(2304) (Show Source): You can put this solution on YOUR website!
What are the factors of 8?
1, 2, 4
.
What are the factors of 15?
1, 3, 5
.
What are the factors of 21?
1, 3, 7
.
The domain set is: {1, 2, 4, 3, 5, 7}
.
What is the least common multiple that is equal to or greater than 21 (the largest number)?
.
1*2*4*3*5*7 = 840
.
840/8 = 105
840/15 = 56
840/21 = 40
.
So, 840 is looking good!
We can check 420, but we'll find it does not work for 8.
420/8 is not an integer solution.
.
But what does the question ask? The smallest 6-digit number that is exactly divisible by 8, 15, and 21.
.
How about 840,000?
.
840,000/8 = 105,000
840,000/15 = 56,000
840,000/21 = 40,000
.
That works, but is it the smallest 6-digit number?
Not likely.
.
Looking back, 8*15*21*x, where x is an integer, will always be divisible exactly by 8, 15 & 21.
We want to find an x such that the product is > 100,000 by the smallest margin.
.
8*15*21=2520
.
2520x >= 100,000
x >= 39.6824397
The next integer is 40.
.
40*2520 = 100,800
.
100,800/8 = 12,600
100,800/15 = 6,720
100,800/21 = 4,800
.
The smallest 6-digit number (that is, a number >= 100,000) that is exactly divisible by 8, 15, and 21 is 100,800.
.
Done.

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