Each of the bases of the powers has three different prime factors:
base
factors power
42
2 3 7 98
70
2 5 7 70
105
3 5 7 42
154
2 7 11 28
The prime factors involved are:
2
3
5
7
11
In the table below, the column at left indicates the unique prime factors involved in a potential divisor, so for example, in the row labeled 2 7, in order to get at least two of the mentioned numbers to be divisible by a number consisting of positive powers of 2 and of 7, they have to divide 70^70, as they will then automatically divide also 42^98. There are 70 possible powers of 2 that can go into this number and 70 possible values of 7, for 4900 possible combinations. The cases of zero power were covered in the lines for 2 and for 7 by themselves (i.e., powers of 2 and of 7), so only powers 1 through 70 are considered.
at least:
1 2 3 4
1 1 1 1 1
2 98 70 28
3 98 42
5 70 42
7 98 70 42 28
11 28
2 3 9604
2 5 4900
2 7 9604 4900 784
2 11 784
3 5 1764
3 7 9604 1764
5 7 4900 1764
7 11 784
2 3 7 941192
2 5 7 343000
3 5 7 74088
2 7 11 21952
Putting these numbers into a spreadsheet leads to the totals:
1,422,569; 8,653; 855 and 29.
Edited on December 17, 2011, 4:14 pm
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Posted by Charlie
on 2011-12-17 12:51:51 |
solution
