Question

Each of the positive integers from 1 to 100 are written on a sheet of paper. Some of these integers are erased. The product of those integers still on the paper leaves a remainder of 4 when divided by 5. Least number of integers that could be erased?

Answer 1

21

Explanation:

You want to know the least number of integers that can be excluded from the product represented by 100! such that the remainder is 4 when divided by 5.

Factors

The factors of 100! will include multiples of 5. All 20 of those must be excluded, otherwise the remainder will always be 0.

The remaining factors, mod 5, will have values of 1, 2, 3, or 4. In the remaining product.

Remainder of 4

Removing one of the factors that has a mod 5 value of 4 will result in the product having a mod 5 value of 4. That's one more number.

The least number of integers that could be erased is 21.

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