Problem 047

[hk]

It states that 644, 645, 646 have three distinct prime factors, that is wrong because 2 is a prime factor of 644 and 646.

Can anybody clarify the problem.

644 = 2² × 7 × 23
645 = 3 × 5 × 43
646 = 2 × 17 × 19
Each of these three numbers has three distinct prime factors.

[meduza2]

Each of these three numbers has three distinct prime factors.

As I said it's wrong, 644 and 646 have a common prime factor. Or your use a non-standard definition for "prime factor".

[hk]

The fact that 644 and 646 have a common prime factor is totally irrelevant.

[sjhillier]

Each of these three numbers has three distinct prime factors.

hk has now answered your question twice, and note he has already answered exactly the same question earlier in the topic, see this post. I'm sure at least one of these answers will help if you think about it carefully.

[meduza2]

Now I understand. But I still think the original text is totally ambiguous. Let $a_1,\ldots,a_4$ be the asked consecutive four numbers, the problem can be understood as:

1) there are set of 16 distinct primes such that every $a_i$ have exactly 4 prime factors from the set, and $a_i$ don't have common prime factors with $a_j$, $i\neq j$.

2) there are set of 4 distinct primes such that every $a_i$ have prime factors from the set, and don't have other prime factors.

3) there are set of 4..16 prime factors, such that every $a_i$ have exactly 4 prime factors from the set, and don't have other prime factors.

4) there are set of 4 distinct primes such that every $a_i$ have exactly 4 prime factors from the set, and don't have other prime factors.

The problem asks... hm... guess what it asks from the original text. As I see frm the topic, not only me find this ambiguous. Do you consider to editing the problem for future solvers?

[hk]

If there is consensus for a better wording that can always be done.
However, the wording should stay short without unnecessary longwindedness.

Would it help if we changed:
"Find the first four consecutive integers to have four distinct prime factors."
into
"Find the first four consecutive integers to have four distinct prime factors each."?

[meduza2]

Advice: between unambiguous and short always choose unambiguous. It's not a telegram, it's a math and programming problem.

Find the first four consecutive integers such that each have exactly 4 distinct prime factors, the integers can have common prime factors.

[hk]

Math learns us: from four consecutive numbers there are always 2 having a factor 2 in common.
So "the integers can have common prime factors" is superfluous and unnecessarily longwinded. This doesn't resolve any ambiguity that cannot be resolved by a little thinking.

[Animus]

Would it help if we changed:
"Find the first four consecutive integers to have four distinct prime factors."
into
"Find the first four consecutive integers to have four distinct prime factors each."?

I think that would be a good idea, making the condition asked for very clear.

[hk]

Done.