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Which one to use:

"I can't remember the number of times I.."

or

"I couldn't remember the nu..."

(Some websites say that both could be used interchangeably.)
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If your memory failure is occurring at the moment you are speaking,

I can't remember the number of times ... [At this moment now, I am unable to remember ...]

If your memory failed you in the past, and you are telling someone about it now,

I couldn't remember the number of times ... [At that moment in the past, I was unable to remember ...]

CJ
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