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by GMATGuruNY » Fri Oct 20, 2017 5:06 am
LUANDATO wrote:What is the greatest prime factor of $$49^{19}-7^{35}$$

A)2
B)3
C)7
D)13
E)19
49¹� - 7³�

= (7²)¹� - 7³�

= 7³� - 7³�

= 7³�(7³ - 1)

= 7³�(343 - 1)

= 7³�(342)

= 7³�(9*38)

= 7³�(3*3*2*19).

The greatest prime factor is the value in blue.

The correct answer is E.
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by EconomistGMATTutor » Fri Oct 20, 2017 9:08 am
What is the greatest prime factor of $$49^{19}-7^{35}$$

A)2
B)3
C)7
D)13
E)19

The OA is E.

Can any expert help me with this PS question please? Thanks.
Hi LUANDATO,
Lets take a look at your question.

$$49^{19}-7^{35}$$
$$=\left(7^2\right)^{19}-7^{35}$$
$$=7^{38}-7^{35}$$
$$=7^{35+3}-7^{35}$$
$$=7^{35}.7^3-7^{35}$$
$$=7^{35}\ \left(7^3-1\right)$$
$$=7^{35}\ \left(343-1\right)$$
$$=7^{35}\ \left(342\right)$$

Write 342 as a product of its prime factors.
$$=7^{35}\ \left(2\times3\times3\times19\right)$$

We can see that the greatest prime factor is 19.
Therefore, Option E is correct.

Hope this helps.
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by Scott@TargetTestPrep » Fri Nov 22, 2019 11:51 am
BTGmoderatorLU wrote:What is the greatest prime factor of $$49^{19}-7^{35}$$

A)2
B)3
C)7
D)13
E)19

The OA is E.

Can any expert help me with this PS question please? Thanks.

Rewriting 49^19 as (7^2)^19 or 7^38, we have:

7^38 - 7^35

7^35(7^3 - 1)

7^35(342)

7^35(19 x 18)

7^35(19 x 2 x 3^2)

Of the four prime factors (7, 19, 2, and 3), we see that 19 is the greatest.

Answer: E

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by Brent@GMATPrepNow » Fri Nov 22, 2019 3:54 pm
BTGmoderatorLU wrote:What is the greatest prime factor of $$49^{19}-7^{35}$$

A)2
B)3
C)7
D)13
E)19

The OA is E.

Can any expert help me with this PS question please? Thanks.
49^19 - 7^35 = (7^2)^19 - 7^35
= 7^38 - 7^35
= (7^35)(7^3 - 1)
= (7^35)(343 - 1)
= (7^35)(342)
= (7^35)(2)(3)(3)(19)

Answer: E

Cheers,
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