Yes you can conceive on your period but it is very unlikely. You are more likely if your periods are frequent and irregular. Less likely if you have regular periods.
The best way to prevent pregnancy at any time is to use a contraceptive.
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You can't conceive on your period full-stop.
Conception refers to when the egg is fertilised by the sperm, which then goes on to implant in the uterus resulting in pregnancy. As there is no egg present during menstruation there is no way that you can conceive during menstruation, it's simply not possible.
You can get pregnant as a result of sex during menstruation, there may be no egg but if your cycle is short you may have fertile quality cervical mucus present - this can keep sperm alive in the vagina for up to a week, by which time you may ovulate and thus conceive. But this wouldn't effect bleeding.