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OLEGan [10]
2 years ago
9

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

415%5E6%20-%207%5E6%24" id="TexFormula1" title="Find the largest prime divisor of $15^6 - 7^6$" alt="Find the largest prime divisor of $15^6 - 7^6$" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
Novosadov [1.4K]2 years ago
7 0

Answer:

Largest prime factor of the given expression is <em>379.</em>

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 <em>379.</em>

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