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N-Divisibility II (Posted on 2011-12-17) Difficulty: 3 of 5
Consider four base ten positive integers 4298, 7070, 10542 and 15428 and, determine the total number of positive integers dividing:

(I) At least one of the four given numbers.

(II) At least two of the four given numbers.

(III) At least three of the four given numbers.

(IV) Each of the four given numbers.

No Solution Yet Submitted by K Sengupta    
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Solution solution | Comment 4 of 5 |

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:





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
  Posted by Charlie on 2011-12-17 12:51:51

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