# Các bài toán số học trong đề thi APMOPS 2010

Question 1: Find the last digit of the number $2\times 2\times 2\times...\times 2$ (859435 of 2’s).

Solution: It is easy to prove by induction that the last digit of the number having the form $2^{4k+3}$ is 8. Hence, the last digit of the number given in the question is 8.

Question 2: Find the value of $1^3+2^3+3^3+4^3+...+20^3+21^3.$

Solution: Remember that for any natural number $n \geq 1$, we always have $1^3+2^3+...+n^3=(\frac{n(n+1)}{2})^2.$

Thus, $1^3+2^3+...+21^3=53361.$

Question 3: Find the smallest whole number that is not a factor of $1\times 2\times 3\times ... \times 21\times 22\times 23.$

Solution: We just need to consider these five numbers: 24, 25, 26, 27, 28, 29 (29 is a prime number).

Notice that 24=4.6 ; 25=5.5 ; 26=2.13 ; 27=3.9 ; 28=4.7 ; Hence 29 is our result.