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.



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