Prime, Composite, Square, and Triangular Numbers

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Questions and Answers

Which statement accurately distinguishes prime and composite numbers?

  • Prime numbers are always even, while composite numbers are always odd.
  • Prime numbers have more than two factors, while composite numbers have exactly two.
  • Prime numbers have exactly two factors, while composite numbers have more than two. (correct)
  • Prime numbers include 1 as a factor, while composite numbers do not.

Why is the number 2 considered a unique prime number?

  • It is the only number that is both prime and a square number.
  • It is the only even prime number. (correct)
  • It is the smallest prime number.
  • It is a factor of all other prime numbers.

Which of the following methods is most efficient for determining if a large number is prime?

  • Checking divisibility by all numbers less than the number.
  • Checking if the number can be expressed as a sum of two square numbers.
  • Checking divisibility by prime numbers less than the square root of the number. (correct)
  • Checking divisibility by all odd numbers less than the number.

Which of the following numbers is a composite number?

<p>33 (C)</p>
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Why is the number 1 neither prime nor composite?

<p>It is only divisible by itself. (D)</p>
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If a number is divisible by 2, what can you conclude about it?

<p>It is a composite number, unless it is 2. (A)</p>
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Which number does not belong to the sequence of square numbers?

<p>75 (D)</p>
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What is the square root of 144?

<p>12 (D)</p>
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Which of the following correctly explains how square numbers are visually represented?

<p>As a square array of dots. (B)</p>
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What is the 7th triangular number?

<p>28 (C)</p>
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What is the formula to calculate the nth triangular number?

<p>$T_n = n * (n + 1) / 2$ (B)</p>
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In a sequence of triangular numbers, what is the difference between consecutive numbers?

<p>It increases by 1 each time. (B)</p>
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Which of the following numbers is both a square number and a triangular number?

<p>36 (D)</p>
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Which option lists numbers that are all prime?

<p>2, 3, 5, 7 (A)</p>
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What is the next triangular number after 10?

<p>15 (C)</p>
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Identify the number that is composite.

<p>15 (B)</p>
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What number is both composite and even?

<p>6 (B)</p>
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Choose the option that contains only square numbers.

<p>1, 4, 9, 16 (B)</p>
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What is the value of the square root of 81?

<p>9 (A)</p>
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If a triangular number is represented by dots arranged in a triangle, what happens to the number of dots in each subsequent row?

<p>It increases by one. (C)</p>
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Flashcards

Prime Number

A whole number greater than 1 that has only two factors: 1 and itself.

Composite Number

A whole number greater than 1 that has more than two factors.

Square Number

A number that can be obtained by multiplying an integer by itself. Also known as a perfect square.

Triangular Number

A number that can be represented as an equilateral triangle of dots.

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Factor

A number that divides another number evenly without leaving a remainder.

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The number 2

The only even prime number.

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How to get a square number?

Multiply an integer by itself.

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Formula for nth triangular number

n * (n + 1) / 2, where n is the position of the number in the sequence.

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Why is '1' not a prime number?

It has only one factor (itself).

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How to check if a large number is prime?

Check for divisibility by prime numbers less than the square root of the number.

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Study Notes

  • Will define and describe prime, composite, square, and triangular numbers.

Prime Numbers

  • A prime number is a whole number greater than 1 that has only two factors (divisors): 1 and itself.
  • A factor is a number that divides another number evenly (without leaving a remainder).
  • Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
  • The number 1 is not considered a prime number, as it has only one factor (itself).
  • The number 2 is the only even prime number. All other even numbers are divisible by 2 and therefore have more than two factors.
  • To determine whether a large number is prime, check for divisibility by prime numbers less than the square root of the number.

Composite Numbers

  • A composite number is a whole number greater than 1 that has more than two factors.
  • A composite number can be divided evenly by numbers other than 1 and itself.
  • Examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
  • The number 1 is neither prime nor composite.
  • All even numbers greater than 2 are composite because they are divisible by 2.
  • To determine whether a number is composite, find any factor other than 1 and itself.

Square Numbers

  • A square number is a number that can be obtained by multiplying an integer by itself.
  • It is also known as a perfect square.
  • Examples of square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
  • 1 = 1 x 1 = 1^2
  • 4 = 2 x 2 = 2^2
  • 9 = 3 x 3 = 3^2
  • 16 = 4 x 4 = 4^2
  • Square numbers can be visually represented as a square array of dots.
  • The square root of a square number is always an integer. For example, the square root of 25 is 5.

Triangular Numbers

  • A triangular number is a number that can be represented as an equilateral triangle of dots where each row contains one more dot than the previous row.
  • The nth triangular number is the sum of the first n natural numbers.
  • The formula for calculating the nth triangular number is: T_n = n * (n + 1) / 2, where n is the position of the number in the sequence.
  • Examples of triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
  • 1 = 1
  • 3 = 1 + 2
  • 6 = 1 + 2 + 3
  • 10 = 1 + 2 + 3 + 4
  • Triangular numbers can be visually represented as a triangle formed by dots.
  • The difference between consecutive triangular numbers increases by 1 each time (2, 3, 4, 5, etc.).

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