Example Question - positive integers
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Finding Natural Numbers with Four Positive Divisors
<p>To find natural numbers with exactly four positive divisors less than 65, we must identify the form of such numbers. A natural number can have exactly four positive divisors if it is in one of the following forms:</p> <p>1. The square of a prime number: \( p^2 \)</p> <p>2. The product of two distinct prime numbers: \( p \times q \)</p> <p>Now, we will find numbers less than 65 for each case.</p> <p>1. For \( p^2 \): The prime numbers less than 8 (since \( 8^2 = 64 \)) are 2, 3, 5, and 7.</p> <p>- \( 2^2 = 4 \)</p> <p>- \( 3^2 = 9 \)</p> <p>- \( 5^2 = 25 \)</p> <p>- \( 7^2 = 49 \)</p> <p>The valid numbers are 4, 9, 25, and 49.</p> <p>2. For \( p \times q \): We consider pairs of primes less than 65:</p> <p>- \( 2 \times 3 = 6 \)</p> <p>- \( 2 \times 5 = 10 \)</p> <p>- \( 2 \times 7 = 14 \)</p> <p>- \( 2 \times 11 = 22 \)</p> <p>- \( 2 \times 13 = 26 \)</p> <p>- \( 2 \times 17 = 34 \)</p> <p>- \( 2 \times 19 = 38 \)</p> <p>- \( 2 \times 23 = 46 \)</p> <p>- \( 2 \times 29 = 58 \)</p> <p>- \( 3 \times 5 = 15 \)</p> <p>- \( 3 \times 7 = 21 \)</p> <p>- \( 3 \times 11 = 33 \)</p> <p>- \( 3 \times 13 = 39 \)</p> <p>- \( 3 \times 17 = 51 \)</p> <p>- \( 3 \times 19 = 57 \)</p> <p>- \( 5 \times 7 = 35 \)</p> <p>- \( 5 \times 11 = 55 \)</p> <p>- \( 5 \times 13 = 65 \) (not valid)</p> <p>- \( 7 \times 11 = 77 \) (not valid)</p> <p>The valid products are 6, 10, 14, 15, 21, 22, 26, 34, 35, 38, 39, 46, 49, 51, 55, 57, and 58.</p> <p>The complete list of natural numbers less than 65 with exactly four positive divisors is: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 34, 35, 38, 39, 49, 51, 55, 57, and 58.</p> <p>Counting these, we have a total of 18 numbers.</p> <p>Thus, the answer is 18.</p>
Question on Identifying a Set of Numbers from a Given Number Line Diagram
The provided image shows a number line with some integers highlighted. To solve the question, we need to list the highlighted integers in the correct order as they appear on the number line from left to right. <p>\{-3, -2, -1, 0, 1, 2, 3\}</p> Based on the options given in the question, the correct set of numbers corresponding to the highlighted integers on the number line is Option B.
Finding Odd Numbers with Provided Digits
To find out how many odd positive integers less than 100 can be represented using the provided digits (2, 3, 8, and 9), we need to look at the possible combinations of these digits that will give us two-digit odd numbers (since all single-digit odd numbers are less than 10 and would not use a combination of the provided digits). An integer is odd if its last digit (ones place) is odd. Among the given digits, the only odd digits are 3 and 9. These digits can only appear in the ones place for the number to be odd. For the tens place, we can use any of the remaining digits (2, 3, 8, and since we've used 9 in ones place, 9 can't be used for two consecutive digits numbers), but we need to ensure that we are not forming a number greater than or equal to 100. So, the possible tens-place digits when the ones place is 3 are 2, 8, and 9 (we can't use 3 at both places as it would form the number 33, which is not greater than any two-digit number). That gives us these numbers: 23, 83, and 93. When the ones place is 9, the possible tens-place digits are 2, 3, and 8, which give us these numbers: 29, 39, and 89. Adding these up, we have: For ones place = 3: 23, 83, 93 (3 numbers) For ones place = 9: 29, 39, 89 (3 numbers) In total, there are 3 + 3 = 6 odd positive integers less than 100 that can be formed using the digits 2, 3, 8, and 9.
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