Question

$explain \: why \: 7 \times 11 \times 13 + 13 \: and \: 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \: are \: composite \: numbers.$
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Answer 1

$\underline\mathtt\green{Answer:}$

• Composite numbers also known as composites in Mathematics are numbers that have more than 2 factors, not like prime numbers that have only one factor, i.e. 1 and the number itself.
• Composite numbers are all natural numbers that are not prime numbers since they can be divided by more than two numbers.
• 6 is composite, for ins,tance, since it is divisible by 1, 2, 3 and even 6.

Now consider,

7 x 11 x 13 + 13

We can also write this as

⇒ (77 + 1) x 13

⇒ 1014

So the factors are

⇒ 13 x 13 x 3 x 2

It is a composite number. The given number has more than two factors.

Now consider

7 x 6 x 5 x 4 x 3 x 2 x 1 + 5

We can also write it as

⇒ 5(7 x 6 x 5 x 4 x 3 x 2 x 1 + 1)

By calculation we get

⇒ 1009 x 5

It is a composite number. The given number has more than two factors.

Hence proved.

Step-by-step explanation:

☺️..

Answer 2

Step-by-step explanation:

Now, we have to explain 7×11×13+137×11×13+13 and 7×6×5×4×3×2×1+57×6×5×4×3×2×1+5 are composite numbers.

So, we have to prove a given number has more than two factors (have factors other than 1 and the number itself).

Now,

⇒7×11×13+13⇒13(7×11+1)⇒13(77+1)⇒13×78⇒13×13×6⇒13×13×3×2⇒7×11×13+13⇒13(7×11+1)⇒13(77+1)⇒13×78⇒13×13×6⇒13×13×3×2

We can see that given numbers have more than two factors (other than 1 and number itself). So, 7×11×13+137×11×13+13 number is a composite number.

Now,

⇒7×6×5×4×3×2×1+5⇒5(7×6×4×3×2×1+1)⇒5(1008+1)⇒5×1009⇒7×6×5×4×3×2×1+5⇒5(7×6×4×3×2×1+1)⇒5(1008+1)⇒5×1009

We can see given numbers have more than two factors (other than 1 and number itself). So, 7×6×5×4×3×2×1+57×6×5×4×3×2×1+5 number is a composite number.