Difference between revisions of "1966 AHSME Problems/Problem 29"
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== Solution == | == Solution == | ||
− | The number of positive integers under <math>1000</math> that are divisible by <math>5</math> is <math>\lfloor\frac{999}{5}\rfloor=199</math>. The number of positive integers under <math>1000</math> that are divisible by <math>7</math> is <math>\lfloor\frac{1000}{7}\rfloor=142</math>. Adding them together, we get <math>341</math>. However, we have over counted the numbers which are divisible by <math>35</math>. There are <math>\lfloor\frac{999}{35}\rfloor=28</math> of these. So, the number of positive integers divisible be <math>7</math> or <math>5</math> under <math>1000</math> is <math>341-28=313</math>. We can conclude that the number of positive integers divisible by neither <math>5</math> or <math>7</math> is <math>999-313=686</math> or answer choice <math>\ | + | The number of positive integers under <math>1000</math> that are divisible by <math>5</math> is <math>\lfloor\frac{999}{5}\rfloor=199</math>. The number of positive integers under <math>1000</math> that are divisible by <math>7</math> is <math>\lfloor\frac{1000}{7}\rfloor=142</math>. Adding them together, we get <math>341</math>. However, we have over counted the numbers which are divisible by <math>35</math>. There are <math>\lfloor\frac{999}{35}\rfloor=28</math> of these. So, the number of positive integers divisible be <math>7</math> or <math>5</math> under <math>1000</math> is <math>341-28=313</math>. We can conclude that the number of positive integers divisible by neither <math>5</math> or <math>7</math> is <math>999-313=686</math> or answer choice <math>\fbox{B}</math>. |
== See also == | == See also == |
Latest revision as of 07:44, 1 January 2024
Problem
The number of positive integers less than divisible by neither nor is:
Solution
The number of positive integers under that are divisible by is . The number of positive integers under that are divisible by is . Adding them together, we get . However, we have over counted the numbers which are divisible by . There are of these. So, the number of positive integers divisible be or under is . We can conclude that the number of positive integers divisible by neither or is or answer choice .
See also
1966 AHSME (Problems • Answer Key • Resources) | ||
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Followed by Problem 30 | |
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