Question

For how many positive integers $n \le 1000$ is $\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$ not divisible by $3?$ (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x.$ )

$22$

$23$

$24$

$25$

$26$

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Written solution:

Write $1000=qn+r$ , where $0\le r<n$ . Then $\left\lfloor\dfrac{1000}{n}\right\rfloor=q$ . The other two floors are usually also $q$ , except that subtracting $1$ or $2$ from $1000$ crosses a multiple of $n$ when $r$ is small.

For $n>1$ , the sum is not divisible by $3$ exactly when $r=0$ or $r=1$ . The case $r=0$ gives divisors of $1000$ , excluding $1$ , for $15$ values. The case $r=1$ gives divisors of $999$ , excluding $1$ , for $(3+1)(1+1)-1=7$ values. The total is $22$ . Thus, A is the correct answer.

2020 AMC 10A Problem 22

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Problem 22 in Other Years