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How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1,2,3, and 6.)

Explanation	Calculations	Help
$We observe that an integer has exactly three positive factors only when it is the square of a prime.$	$n = p^{2} where p is prime$	Theory & Tactics Method Card DIV9-B Click to view full details
$We determine all prime values of p for which p² is at most 16, by square rooting the inequality$	$p^{2} \leq 16 p \leq 4 primes p = 2, 3$	Theory & Tactics Method Card INEQ1-E Click to view full details
$We list the squares of these primes and count how many lie between 1 and 16.$	$2^{2} = 4 3^{2} = 9 number of such integers = 2$

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How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1,2,3, and 6.)

Explanation	Calculations	Help
$We observe that an integer has exactly three positive factors only when it is the square of a prime.$	$n = p^{2} where p is prime$	Theory & Tactics Method Card DIV9-B Click to view full details
$We determine all prime values of p for which p² is at most 16, by square rooting the inequality$	$p^{2} \leq 16 p \leq 4 primes p = 2, 3$	Theory & Tactics Method Card INEQ1-E Click to view full details
$We list the squares of these primes and count how many lie between 1 and 16.$	$2^{2} = 4 3^{2} = 9 number of such integers = 2$

Scroll horizontally to view all columns

2