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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

Rating: 5.0000 (1 votes)

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Third things second (partial spoiler)

| Comment 2 of 5 |

The four numbers are (2*3*7)^98, (2*5*7)^70, (3*5*7)^42, and (2*7*11)^28.

Part (III) --

Any number which divides the first three must divide 7^42. There are 43 of these.

Any number which divides the last three must divide 7^28. This is a subset of the 43 already counted.

Any number which divides the 1st, 3rd and 4th must divide 7^28. This is a subset of the 43 already counted.

Finally, any number which divides the 1st, 2nd and 4th must divide (2*7)^28. The first 43 includes all of those which are not a power of 2, so we have 28 powers of 2 * 29 powers of 7 = 812 numbers not included in the 43.

Total which divide at least 3 are 43 + 812 = 855.

Posted by Steve Herman on 2011-12-17 12:28:44

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