There are two types of numbers, Prime Numbers and Composite Numbers.
The Prime Numbers between 1 - 100 are
1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97. Since there is no common factor between these numbers, to find the least number which is divisible by all above numbers, we will have to take product of these numbers.
Hence part one of our answer
x = 1*2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59*61*67*71*73*79*83*89*97
The composite numbers between 1-100 are
4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,
49,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,74,75,76,77,78,80,81,82,84,85,86,
87,88,90,91,92,93,94,95,96,98,99,100
So let's try to find the smallest number which along with x will be divisible by all the above composite numbers.
4,6 = 4 (2,3 From x along with 2 from 4 will give us 12)
4,8 = 4 (2,3 from x along with 4 gives us 24)
4,9 = 12 (2,3 from x along with 2 from 4 will still need 3 to give us 36)
12,10 = 12 (2,5, from x gives us 10)
12,12 = 12
12,14 = 12 (7 from x along with 2 from 12 gives us 14)
12,15 = 12 (3,5 from x takes care of 15)
12,16 = 24 (2 from x and 4 from 12 will still need additional 2 to make it multiple of 16)
24,18 = 24 (3 from x and 24 gives 72)
24,20 = 24 (2,5 from x and 2 from 24 gives multiple of 20)
24,21 = 24 (3,5 from x takes care of 15)
24,22 = 24 (2,11 from x takes care of 22)
24,24 = 24
24,25 = 120 (5 from x needs additional 5)
120,26 = 120 (2,13 from x takes care of 22)
120,27 = 360 (3 from x, 3 from 120 needs additional 3 for 27)
360,28 = 360 (7 from x and 4 from 360 forms 28)
360,30 = 360
360,32 = 720 (2 from x, 8 from 360 needs additional 2 for 32)
720,33 = 720 (3,11 from x takes care of 33)
720,34 = 720 (2,17 from x takes care of 34)
720,35 = 720 (5,7 from x takes care of 35)
720,36= 720
720,38= 720 (2,19 from x takes care of 38)
720,39= 720 (3,13 from x takes care of 39)
720,40= 720
720,42 = 720 (2,3,7 from x takes care of 42)
720,44 = 720 (2,11 from x takes care of 22, supplemented by 2 from 720)
720,45 = 720
720,46 = 720 (2,23 from x takes care of 22)
720,48 = 720
720,49 = 5040 (7 from x needs additional 7)
5040, 50 = 5040 (5 from x and 10 from 5040 will form 50)
5040, 51 = 5040 (3,17 from x takes care of 51)
5040, 52 = 5040 (13 from x supplemented by 4 from 5040 takes care of 52)
5040, 54 = 5040
5040, 55 = 5040 (11 from x and 5 from 5040 forms 55)
5040, 56 = 5040 (7 from x and 8 from 5040 forms 56)
5040, 57 = 5040 (3, 19 from x forms 57)
5040, 58 = 5040 (2, 29 from x forms 58)
5040, 60 = 5040
5040, 62 = 5040 (2,31 from x forms 62)
5040, 63 = 5040 (3,7 from x supplemented by 3 from 5040 gives 63)
5040, 64 = 10080 (2 from x, 16 from 5040 needs additional 2)
10080, 65 = 10080 (5,13 from x forms 65)
10080, 66 = 10080 (2,3,11 from x forms 66)
10080, 68 = 10080 (2,17 from x along with 2 from 10080 forms 68)
10080, 69 = 10080 (3, 23 from x forms 69)
10080, 70 = 10080 (2,5,7 from x forms 70)
10080, 72 = 10080
10080, 74 = 10080 (2, 37 from x forms 74)
10080, 75 = 10080 (3,5 from x along with 5 from 10080 forms 75)
10080, 76 = 10080 (2, 19 from x along with 2 from 10080 forms 76)10080, 77 = 10080 (7, 11 from x forms 77)
10080, 78 = 10080 (2, 3, 13 from x forms 78)
10080, 80 = 10080
10080, 81 = 30240 (3 from x, 9 from 10080 needs additional 3)
30240, 82 = 30240 (2, 41 from x forms 82)
30240 , 84 = 30240 (2,3,7 from x along with 2 from 30240 forms 84)
30240 , 85 = 30240 (5,17 from x forms 85)
30240 , 86 = 30240 (2,43 from x forms 86)
30240 , 87 = 30240 (3,29 from x forms 87)
30240 , 88 = 30240 (2,11 from x and 4 from 30240 forms 88)
30240 , 90 = 30240 (3 from x and 30 from 30240 forms 90)
30240 , 91 = 30240 (7, 13 from x forms 91)
30240 , 92 = 30240 (2, 23 from x along with 2 from 30240 forms 92)
30240 , 93 = 30240 (3, 31 from x forms 93)
30240 , 94 = 30240 (2, 47 from x forms 94)30240 , 95 = 30240 (5, 19 from x forms 95)30240 , 96 = 30240
30240 , 98 = 30240 (7 from x and 14 from 30240 forms 98)
30240 , 99 = 30240 (3,11 from x and 3 from 30240 forms 99)
30240 , 100 = 30240 (2,5 from x and 10 from 30240 gives 100)
Hence the final answer is
= 1*2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59*61*67*71*73*79*83*89*97*30240
= 6.9720375229712477164533808935312 *10^40
Let me know if you find errors!
The same calculation can be used to find smallest number divisible by any range up to 100.
Say the range is 1-10.
The Prime numbers to be considered are 1*2*3*5*7
The composite numbers calculation till 10 is
4,6 = 6 (2 From x along with 6 will give us 12)
6,8 = 12 (2,3 from x along with 2 from 8 gives us 12)
12,9 = 12 (3 from x along with 3 from 12 gives us 9)
12,10 = 12 (2,5, from x gives us 10)
So answer is = 1*2*3*5*7*12 = 2520.
The Prime Numbers between 1 - 100 are
1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97. Since there is no common factor between these numbers, to find the least number which is divisible by all above numbers, we will have to take product of these numbers.
Hence part one of our answer
x = 1*2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59*61*67*71*73*79*83*89*97
The composite numbers between 1-100 are
4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,
49,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,74,75,76,77,78,80,81,82,84,85,86,
87,88,90,91,92,93,94,95,96,98,99,100
So let's try to find the smallest number which along with x will be divisible by all the above composite numbers.
4,6 = 4 (2,3 From x along with 2 from 4 will give us 12)
4,8 = 4 (2,3 from x along with 4 gives us 24)
4,9 = 12 (2,3 from x along with 2 from 4 will still need 3 to give us 36)
12,10 = 12 (2,5, from x gives us 10)
12,12 = 12
12,14 = 12 (7 from x along with 2 from 12 gives us 14)
12,15 = 12 (3,5 from x takes care of 15)
12,16 = 24 (2 from x and 4 from 12 will still need additional 2 to make it multiple of 16)
24,18 = 24 (3 from x and 24 gives 72)
24,20 = 24 (2,5 from x and 2 from 24 gives multiple of 20)
24,21 = 24 (3,5 from x takes care of 15)
24,22 = 24 (2,11 from x takes care of 22)
24,24 = 24
24,25 = 120 (5 from x needs additional 5)
120,26 = 120 (2,13 from x takes care of 22)
120,27 = 360 (3 from x, 3 from 120 needs additional 3 for 27)
360,28 = 360 (7 from x and 4 from 360 forms 28)
360,30 = 360
360,32 = 720 (2 from x, 8 from 360 needs additional 2 for 32)
720,33 = 720 (3,11 from x takes care of 33)
720,34 = 720 (2,17 from x takes care of 34)
720,35 = 720 (5,7 from x takes care of 35)
720,36= 720
720,38= 720 (2,19 from x takes care of 38)
720,39= 720 (3,13 from x takes care of 39)
720,40= 720
720,42 = 720 (2,3,7 from x takes care of 42)
720,44 = 720 (2,11 from x takes care of 22, supplemented by 2 from 720)
720,45 = 720
720,46 = 720 (2,23 from x takes care of 22)
720,48 = 720
720,49 = 5040 (7 from x needs additional 7)
5040, 50 = 5040 (5 from x and 10 from 5040 will form 50)
5040, 51 = 5040 (3,17 from x takes care of 51)
5040, 52 = 5040 (13 from x supplemented by 4 from 5040 takes care of 52)
5040, 54 = 5040
5040, 55 = 5040 (11 from x and 5 from 5040 forms 55)
5040, 56 = 5040 (7 from x and 8 from 5040 forms 56)
5040, 57 = 5040 (3, 19 from x forms 57)
5040, 58 = 5040 (2, 29 from x forms 58)
5040, 60 = 5040
5040, 62 = 5040 (2,31 from x forms 62)
5040, 63 = 5040 (3,7 from x supplemented by 3 from 5040 gives 63)
5040, 64 = 10080 (2 from x, 16 from 5040 needs additional 2)
10080, 65 = 10080 (5,13 from x forms 65)
10080, 66 = 10080 (2,3,11 from x forms 66)
10080, 68 = 10080 (2,17 from x along with 2 from 10080 forms 68)
10080, 69 = 10080 (3, 23 from x forms 69)
10080, 70 = 10080 (2,5,7 from x forms 70)
10080, 72 = 10080
10080, 74 = 10080 (2, 37 from x forms 74)
10080, 75 = 10080 (3,5 from x along with 5 from 10080 forms 75)
10080, 76 = 10080 (2, 19 from x along with 2 from 10080 forms 76)10080, 77 = 10080 (7, 11 from x forms 77)
10080, 78 = 10080 (2, 3, 13 from x forms 78)
10080, 80 = 10080
10080, 81 = 30240 (3 from x, 9 from 10080 needs additional 3)
30240, 82 = 30240 (2, 41 from x forms 82)
30240 , 84 = 30240 (2,3,7 from x along with 2 from 30240 forms 84)
30240 , 85 = 30240 (5,17 from x forms 85)
30240 , 86 = 30240 (2,43 from x forms 86)
30240 , 87 = 30240 (3,29 from x forms 87)
30240 , 88 = 30240 (2,11 from x and 4 from 30240 forms 88)
30240 , 90 = 30240 (3 from x and 30 from 30240 forms 90)
30240 , 91 = 30240 (7, 13 from x forms 91)
30240 , 92 = 30240 (2, 23 from x along with 2 from 30240 forms 92)
30240 , 93 = 30240 (3, 31 from x forms 93)
30240 , 94 = 30240 (2, 47 from x forms 94)30240 , 95 = 30240 (5, 19 from x forms 95)30240 , 96 = 30240
30240 , 98 = 30240 (7 from x and 14 from 30240 forms 98)
30240 , 99 = 30240 (3,11 from x and 3 from 30240 forms 99)
30240 , 100 = 30240 (2,5 from x and 10 from 30240 gives 100)
Hence the final answer is
= 1*2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59*61*67*71*73*79*83*89*97*30240
= 6.9720375229712477164533808935312 *10^40
Let me know if you find errors!
The same calculation can be used to find smallest number divisible by any range up to 100.
Say the range is 1-10.
The Prime numbers to be considered are 1*2*3*5*7
The composite numbers calculation till 10 is
4,6 = 6 (2 From x along with 6 will give us 12)
6,8 = 12 (2,3 from x along with 2 from 8 gives us 12)
12,9 = 12 (3 from x along with 3 from 12 gives us 9)
12,10 = 12 (2,5, from x gives us 10)
So answer is = 1*2*3*5*7*12 = 2520.
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