Figure 1
Answer: The “third number” common to both sums was 93.
The winner is Martin Kipps of Wokingham, Berkshire, UK. There were 192 entries.
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Worked answer
2-digit numbers that are the product of two different primes are shown in Figure 1.
Since the summands have no common factors they cannot both be even.
In each sum the smaller summand must have its tens digit less than 5 to enable the sum to be a two-digit number.
Since there is only one eligible number whose tens digit is 4, that tens digit must be 1, 2 or 3. The larger summand cannot have tens digit 9.
For 15 and 14, a parallel couple is 35 and 34: 15 + 34 = 14 + 35; but the latter pair have a factor in common (7).
There are no parallel couples to 21 and 22, or 21 and 26.
There is no number after 33 whose unit digit is 3 before 93. There are no parallel couples (other than one whose tens digit is 9) for 35 and 34, 39 and 34, or 39 and 38. 35 and 39 have a parallel couple in 65 and 69, but 35 + 69 = 104, a 3-digit number (which is not the product of 2 primes).
That leaves 35 and 38, with 55 and 58:
35 (5 x 7) + 58 (2 x 29) = 38 (2 x 19) + 55 (5 x 11) = 93 (3 x 31)
