Question 6 of 10intermediate🔍 AnalyzeLong Answer4 marks

Explain why 7×11×13+137 \times 11 \times 13 + 13 and 7×6×5×4×3×2×1+57 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 are composite numbers.

Correct Answer

First number: 7×11×13+137 \times 11 \times 13 + 13

=13×(7×11+1)= 13 \times (7 \times 11 + 1) =13×(77+1)= 13 \times (77 + 1) =13×78= 13 \times 78 =13×2×3×13= 13 \times 2 \times 3 \times 13 =2×3×132= 2 \times 3 \times 13^2

Since this number can be expressed as a product of primes (more than one prime factor), it is a composite number.

Second number: 7×6×5×4×3×2×1+57 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5

=5040+5= 5040 + 5 =5045= 5045 =5×1009= 5 \times 1009

Since this number can be expressed as a product of two numbers greater than 1 (5 and 1009), it is a composite number.

Conclusion: Both numbers have factors other than 1 and themselves, hence both are composite numbers.

Exercise: Exercise 1.1 | Q: 6 | (Page Page 5)
For More Understanding

Explanation

A composite number is a positive integer that has at least one factor other than 1 and itself. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes.

For the first expression, factoring out 13 gives 13(77+1)=13×7813(77 + 1) = 13 \times 78, which clearly has factors 13 and 78. For the second expression, 7×6×5×4×3×2×1=7!=50407 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 7! = 5040, and 5040+5=5045=5×10095040 + 5 = 5045 = 5 \times 1009, which is also composite.

Solution Steps

  1. Step 1: Simplify first expression:

    7×11×13+13=13(77+1)=13×787 \times 11 \times 13 + 13 = 13(77 + 1) = 13 \times 78

  2. Step 2: Factorise 78=2×3×1378 = 2 \times 3 \times 13, so first number:

    =2×3×132= 2 \times 3 \times 13^2

  3. Step 3: Since it has prime factors other than itself, it is composite.

  4. Step 4: Simplify second expression:

    7×6×5×4×3×2×1=50407 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040

  5. Step 5: Calculate:

    5040+5=5045=5×10095040 + 5 = 5045 = 5 \times 1009

  6. Step 6: Since it has factors 5 and 1009, it is composite.

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