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The phrase "a whole number of problems" is correct and usable in written English.
It can be used when referring to a specific quantity of problems that is an integer, emphasizing that the number is complete and not fractional.
Example: "The project encountered a whole number of problems that needed to be addressed before moving forward."
Alternatives: "a complete set of problems" or "an integer number of problems".
Exact(1)
The normalisation of the situation with Iran makes it possible to resolve "a whole number of problems and conflicts in the region" and will have a "positive influence on the situation as a whole", Lavrov said.
Similar(59)
"It's ironic in a whole number of ways".
For small values that are within a tenth value away from a whole number, this does not present a problem.
Unpaired Number of unpaired electrons: a whole number.
Other than the terrible economy and prospect for magazines as a whole, GFW faced a number of problems.
The college as a whole numbers 227.
"This is going to solve a huge number of problems for a great number of people".
It solves a whole load of problems.
It's a whole bundle of problems.
"And that brings with it a whole host of problems.
I think he had a whole lot of problems.
More suggestions(3)
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com