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 N-Divisibility II (Posted on 2011-12-17)
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 No Rating

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 2nd things 3rd (partial spoiler) | Comment 3 of 5 |
The four numbers are (2*3*7)^98, (2*5*7)^70, (3*5*7)^42, and (2*7*11)^28.

Part (II) --
a) Any number which divides the 1st and 2nd must divide (2*7)^70.

b) Any number which divides the 1st and 3rd must divide (3*7)^42.

c) Any number which divides the 1st and 4th must divide (2*7)^28.

d) Any number which divides the 2nd and 3rd must divide (5*7)^42.

e) Any number which divides the 2nd and 4th must divide (2*7)^28.

f) Any number which divides the 3rd and 4th must divide 7^28.

We can ignore cases c, e and f, because they are all subsets of a.

So, we are looking for all numbers which divide (2*7)^70 or (3*7)^42 or (5*7)^42.

To compute these,
1) The number which divide 7^70 (ie, 71)
2) The number which divide (2*7)^70 but which are not a pure power of 7 = (70 powers of 2) * (71 powers of 7) = 4970
3) The number which divide (3*7)^42  but which are not a pure power of 7 = (42 powers of 3) * (43 powers of 7) = 1806
4) The number which divide (5*7)^42  but which are not a pure power of 7 = (42 powers of 5) * (43 powers of 7) = 1806

Total number = 71 + 4970 + 1806 + 1806 = 8653

No time now to solve the 1st part.  Perhaps somebody else will check my work, and finish the problem

 Posted by Steve Herman on 2011-12-17 12:47:51

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