Explain why \(7 × 11 × 13 + 13\) and \(7 × 6 × 5 × 4 × 3 × 2 × 1 + 5\) are composite numbers.
Solution and Explanation
Numbers are of two types - prime and composite. Prime numbers can be divided by \(1\) and only itself, whereas composite numbers have factors other than \(1\) and itself.
It can be observed that
\(7 × 11 × 13 + 13 = 13 × (7 × 11 + 1)\)
\(= 13 × (77 + 1) = 13 × 78\)
\(= 13 ×13 × 6\)
The given expression has \(6\) and \(13\) as its factors. Therefore, it is a composite number.
\(7 × 6 × 5 × 4 × 3 × 2 × 1 + 5\)
\(= 5 ×(7 × 6 × 4 × 3 × 2 × 1 + 1)\)
\(= 5 × (1008 + 1)\)
\(= 5 ×1009\)
Since, \(1009\) cannot be factorised further.
Therefore, the given expression has \(5\) and \(1009\) as its factors. Hence, it is a composite number.
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Concepts Used:
Real Numbers
Real numbers are the set of numbers that includes all rational and irrational numbers. Rational numbers are those that can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as a ratio of integers and have non-repeating, non-terminating decimal expansions. Real numbers also include integers, which are whole numbers and their negative counterparts.
The set of real numbers is represented by the symbol R, and it is an infinite set that includes all possible numbers. It can be visualized as a number line, with negative numbers to the left of zero and positive numbers to the right.
Real numbers are used to represent quantities that can be measured or counted, such as time, distance, and temperature. They are essential in various fields such as science, engineering, economics, and finance.
Read More: Operations on Real Numbers
Real numbers have several properties that make them unique. They are closed under addition, subtraction, multiplication, and division, meaning that when two real numbers are combined using any of these operations, the result is always a real number. Real numbers also have the properties of associativity, commutativity, and distributivity, which help simplify mathematical operations.
Real numbers are an important concept in mathematics, and their properties and relationships with other number sets are studied extensively in algebra, calculus, and other branches of mathematics.