Difference between revisions of "2020 AMC 10A Problems/Problem 22"
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For how many positive integers <math>n \le 1000</math> is<cmath>\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor</cmath>not divisible by <math>3</math>? (Recall that <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.) | For how many positive integers <math>n \le 1000</math> is<cmath>\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor</cmath>not divisible by <math>3</math>? (Recall that <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.) | ||
Revision as of 18:24, 1 February 2020
Contents
[hide]Problem
For how many positive integers is
not divisible by
? (Recall that
is the greatest integer less than or equal to
.)
Solution 1
Let . If the expression is not divisible by
, then the three terms in the expression must be
, which would imply that
is a divisor of
but not
, or
, which would imply that
is a divisor of
but not
.
has
factors, and
has
factors. However,
does not work because
a divisor of both
and
, and since
is counted twice, the answer is
.
Solution 2
Let . Notice that if
is divisible by
, then the three terms in the expression must be
, if
is divisible by
, then the three terms in the expression must be
, and if if
is divisible by
, then the three terms in the expression must be
.
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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