Question

Find the largest prime divisor of $15^6 - 7^6$$Find the largest prime divisor of 15^6 - 7^6$

Answer 1

Answer:

Largest prime factor of the given expression is 379.

Step-by-step explanation:

Given the expression:

$15^6 - 7^6$

To find:

The largest prime factor of the given expression.

Solution:

First of all, let us factorize the given expression.

$15^6 - 7^6\\\Rightarrow (15^3)^2 - (7^3)^2\\\\\text{Using } x^{2} -y^2 = (x+y) (x-y)\\\\\Rightarrow (15^3+7^3)(15^3-7^3)$

Let us learn two formula:

$x^3+y^3 = (x+y)(x^2+y^2-xy)$

$x^3-y^3 = (x-y)(x^2+y^2+xy)$

Applying the above formula in the expression written above:

$(15^3+7^3) = (15+7)(15^2+7^2-15\times 7)\\\Rightarrow 22(225+49-105 ) = 2 \times 11 \times 169 \\\Rightarrow 2 \times 11\times 13\times 13\\\Rightarrow 2 \times 11\times 13^2$ ...... (1)

Similarly:

$(15^3-7^3) = (15-7)(15^2+7^2+15\times 7)\\\Rightarrow 8(225+49+105 ) = 2^3 \times 379$........ (2)

Multiplying the expressions from (1) and (2) to get the result:

$\therefore 15^6 - 7^6 = 2^4 \times 11 \times \underline{\bold{379}} \times 13 ^2$

So, largest prime factor of the given expression is 379.

Answer 2

The largest prime divisor of the expression is 379.

How to solve the divisor?

The expression will be factorized. This will be:

= (15³)² - (7³)²

= (15 + 7)³(15 - 7)³

This can be illustrated as:

(15 + 7)³ = 2 × 11 × 13²

(15 - 7)³ = 2³ × 379

In conclusion, the largest prime divisor of the expression is 379.

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