Confused on comma usage

[JohnDoe12345]

Which of the following sentences, if any, are grammatically correct?

Find the prime factors, which, when multiplied by each other, yield 864.

Find the prime factors which, when multiplied by each other, yield 864.

Find the prime factors, which when multiplied by each other, yield 864.

Find the prime factors, which when multiplied by each other yield 864.

Find the prime factors which when multiplied by each other yield 864.

I would say the last sentence is the only correct one, but when I speak the sentence aloud, there is a clear pause after which.

[5jj]

Start with this:

Find the prime factors which yield 864.

This can stand alone (which means that 'when multiplied by each other' merely gives extra information). It also means that 'which yield 864' is a defining relative clause. There is no pause between 'factors' and 'which', and 'that' can replace 'which'.

So, there is no comma between 'factors' and 'which'.

We have seen that the 'when' clause gives extra information. This information is bracketed off by slight pauses in speech; we can do that in writing with commas. SO:.

2. Find the prime factors which, when multiplied by each other, yield 864.

[JohnDoe12345]

How about the following:

Find two prime numbers which when multiplied by each other equal 864.

In this case, "when multiplied by each other" is essential to the sentence, so it cannot be surrounded by commas, right?

[5jj]

Find two prime numbers which when multiplied by each other equal 864.

In this case, "when multiplied by each other" is essential to the sentence, so it cannot be surrounded by commas, right?

They may be essential to the mathematician, not to the speaker. The speaker will normally make a pause befor 'when' and after 'other'.

The following is a perfectly grammatical sentence in English:

Find two prime numbers which equal 864.

Your original appears quite natural as:

Find two prime numbers which (when multiplied by each other) equal 864.