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Semiprime Switch (Posted on 2011-01-18)

Tom, Dick and Harry each chose two different 2-digit semiprime numbers (a semiprime is the product of exactly two different prime numbers). Any given semiprime may have been used by more than one of these characters, but any one person chose two different ones. In each of the three cases, the sum of the two semiprimes was a 3-digit semiprime.

Also, in each instance, if the units digits of the two semiprimes were swapped, to be paired with the other tens digit, the result was two semiprimes that were different from the original pair. Of course they added up to the same 3-digit semiprime as the first pair. In each of Tom's, Dick's and Harry's three semiprimes, none of the six primes from which they were made were the same: for example, if Tom's semiprimes were 93 and 94 adding up to 187, all six primes (3 and 31 for 93, 2 and 47 for 94 and 11 and 17 for 187) would all be different, satisfying this rule. The same was also true for each of the three people after the swap of units digits between the two 2-digit numbers.

Tom's, Dick's and Harry's 3-digit sums were all different, but Tom's and Harry's 3-digit sums did share a common prime factor, while Dick's did not.

What were Dick's numbers?
What were the other two sets of numbers?

Submitted by Charlie

Rating: 5.0000 (1 votes)

Solution: (Hide)

Here's a list of the pairs of 2-digit semiprimes that add up to a 3-digit semiprime, where none of the six primes involved in their makeup are the same.
In only three instances are such cases related where swapping the pairing of unit with tens places results in the same situation, and those are marked with asterisks.
Two of the cases involve the prime 2 in the 3-digit sum, and are therefore Tom's and Harry's. Dick's set is 35 and 94 (or 34 and 95) adding to 129.

93 94 187 91 94 185 85 93 178 87 91 178 82 95 177 86 91 177 74 87 161 74 85 159 65 94 159 77 82 159 65 93 158 69 86 155 51 95 146 * 2*73 55 91 146 * 69 77 146 51 94 145 57 86 143 58 85 143 69 74 143 55 87 142 * 2*71 57 85 142 * 51 91 142 65 77 142 55 86 141 46 95 141 39 95 134 65 69 134 57 77 134 46 87 133 39 94 133 51 82 133 35 94 129 * 3*43 34 95 129 * 38 91 129 55 74 129 38 85 123 58 65 123 46 77 123 35 87 122 57 65 122 26 93 119 33 86 119 57 62 119 33 85 118 33 82 115 57 58 115 21 94 115 38 77 115 22 93 115 26 85 111 46 65 111 34 77 111 21 85 106 15 91 106 51 55 106

From Enigma No. 1618, "No common factors", by Richard England, New Scientist, 23 October 2010, page 28.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): Problem	hoodat	2011-01-21 19:00:49
re: Problem	brianjn	2011-01-20 20:51:12
From the Spreadsheet	brianjn	2011-01-20 20:38:57
Problem	hoodat	2011-01-20 16:23:13
Towards a Spreadsheet	brianjn	2011-01-20 02:21:17

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