The poker card Qs

From a stack of poker cards, the cards A, 2, 3, 4, 5, 6, 7, 8, 9, 10 of spades and A, 2, 3, 4, 5, 6, 7, 8, 9, 10 of Hearts were selected.

Out of these twenty cards, how many ways can one select two cards-one from Hearts and one from spades-such that the product of the two numbers leaves a remainder of 1 when divided by 10?

Easy Math Editor

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Comments

A two-way table as such can be made. With each entry entered as product and then of $\mod {(10)}$ . There are only 4 entries of 1

Mahdi Raza - 1 year ago

WOW thanks

Vicky Clearwater - 1 year ago

We must solve $xy \equiv 1$ mod 10.

For any $x$ that is not a zero divisor modulo 10 (i.e. not a multiple of 2 or 5), this equation has a unique solution modulo 10. For other $x$ , there is no solution. Therefore there are $\phi(10) = 4$ solutions.