Question

state the fundamental theorem of arithmetic and thus show that (2×3×5×7×11+11) is a composite number.​

Answer 1

The fundamental theorem of arithmetic states that every composite number can be expressed as a product of primes and this factorisation is unique for every number.

Looking at this question we have:

2×3×5×7×11+11 = 11{2 × 3 × 5 × 7 +1}

→ 11{210 + 1}

→ 11 × 211

211 cannot be factorised further. Therefore, the given expression has 11 and 211 as its factor. Hence, the given number is composite.

Answer 2

Given :-

(2 × 3 × 5 × 7 × 11 + 11)

To Find :-

Fundamental theorem

Solution :-

Fundamental theorem as defined as that any number except 1 is either prime number or can be broken in prime number.

$\sf 2 \times3\times5\times7\times 11+11$

Taking 11 as common

$\sf 11(2\times3\times 5 \times7+1)$

$\sf 11(6 \times 35 + 1)$

$\sf 11(210 + 1)$

$\sf 11(211)$

211 is a prime number and cannot be broken