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I’d say (for want of a brainstorm) that the usage is idiomatic: ‘pay in cash’; ‘pay by check’; ‘pay by credit card’. '‘Pay with a check/with my credit card/with cash’.
A googling, out of curiosity:
176,000 English pages for “pay with cash”.
612,000 English pages for “pay in cash”
275,000 English pages for “pay by cash”
1,220,000 English pages for “pay by check”
1,430 English pages for “pay in check”
Whatever else holds true, evidently the check is the more popular instrument!
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How many of these Google results were on pages from English-speaking countries, and how many were written by non-native speakers? “Pay by cash” sounds like foreigner English to my fallible ears.
I don’t want to pursue a dispute of the euro issue or even accuse you of deluding yourself, but if you used the euro in the UK, I assume you wouldn’t be too euphoric (view it as a pure euphemism) at first, as you’d need to get used to the unusual, unique, uniform, unifying, singular, not too alluring and sometimes abused currency unit of the European Union (phew!).
My immune view on the soluble issue might be inaccurate and refutable, but it hopefully won’t raise an acute hue and cry in unison.