Problem 11 · The last digit of the product
Four consecutive integers with no multiple of 5: endings 1234 or 6789.
Integer answer, at most 4 digitsJaume multiplies 4 consecutive positive integers, all greater than 2026, and notices that the result is not a multiple of 10. What can the last digit of this result be? Give all possible digits in increasing order. (For example, if the last digit could be 2, 3 or 7, answer 237.)
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: among four consecutive integers there are always two even ones → the product is even. For it not to be a multiple of $10$, none of the four can be a multiple of $5$.
Avoiding multiples of $5$ forces the last digits of the four numbers to be $1, 2, 3, 4$ or $6, 7, 8, 9$. In both cases, the product's last digit:
$$1 \cdot 2 \cdot 3 \cdot 4 = 24 \to 4, \qquad 6 \cdot 7 \cdot 8 \cdot 9 = 3024 \to 4.$$
The only possible final digit is $4$.
Answer: 4