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Change the type of thread
For Eg.:
1. Every paycheck is right. (Correct)
2. Every paychecks are right. (Incorrect.)

What about these? Are they correct? If not, why? What do they mean?
3. I have every ten paycheck from last year.
4. I have every ten paychecks from last year.

5. Every ten car is good.
6. Every ten cars are good.

7. We do this every ten minute.
8. We do this every ten minutes.

Is this incorrect?
9. I have every a ten paycheck from last year
Every has a couple of meanings here.

A) to include all members of a set without exception. This is the use in your sentences 1,2,5 and 6. This use can only be followed by a singular noun. I assume you mean tenth car though rather than ten - this means that cars 1-9 are no good, but car 10 is good.

B) to indicate something happening at specific intervals. This is your useage in sentences 7 and 8. This can be followed by a singular (i.e. one minute) or plural (ten minutes) depending on whether the frequency of event is 'one' or 'other number'.

I cannot follow the meaning of sentences 3, 4 and 9. The inclusion of the word ten (even if interpreted as tenth) is causing me a problem. Do you mean tenth? In that case sentence 3 is correct. I still can't fathom sentence 9 though. 'A' does not make any sense.
I agree with Nona that you have to use "tenth" in 3 and 5. Although this form isn't used nearly as often as "...every ten minutes...", use "tenth" in 7 to make it grammatical. Assuming you got no more than 10 paychecks last year, use "all" instead of "every" in 4. Regarding 3, we have to assume that you got at least 30 paychecks. 9 is also beyond me.

Are these correct? If not, why? What do they mean?
1. We do this every tenth minute.
2. We do this every tenth minutes. (If this is incorrect, why? Is it because 'tenth' is an adjective here? So 'minutes' needs to be 'minute'?)
3. We do this every ten minutes.
I think all but 2 are correct
1. Correct. It means that at every 10th minute mark, you (we) do this. Very similar in meaning to 3.

2. Not ok. The singular must be used but not because 10th is an adjective (numbers are adjectives when modifying a noun). It's because 10th refers to only a single minute - the 10th. So if someone asks, "When do we this?" the reply may be "At the 10th minute, and every 10th minute thereafter.".

3. Correct. The meaning is exactly that, every 10 minutes you do something. The singular for minute isn't used because the 10 refers to how many minutes pass before you (we) do something. Compare this to the explanation in 2.