What is the greatest prime factor...

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BTGmoderatorLU: Moderator; Posts: 2575; Joined: Sun Oct 15, 2017 1:50 pmFollowed by: 6 members

by BTGmoderatorLU » Thu Oct 19, 2017 7:02 pm

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.

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Source: Beat The GMAT — Problem Solving | Thu Oct 19, 2017 7:02 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

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

49Â¹â�¹ - 7Â³â�µ

= (7Â²)Â¹â�¹ - 7Â³â�µ

= 7Â³â�¸ - 7Â³â�µ

= 7Â³â�µ(7Â³ - 1)

= 7Â³â�µ(343 - 1)

= 7Â³â�µ(342)

= 7Â³â�µ(9*38)

= 7Â³â�µ(3*3*2*19).

The greatest prime factor is the value in blue.

The correct answer is E.

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

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.
I am available if you'd like any follow up.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8177; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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