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# What is the greatest prime factor of 422+752+6300

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Manager
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What is the greatest prime factor of 422+752+6300  [#permalink]

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12 Mar 2015, 12:08
4
18
00:00

Difficulty:

95% (hard)

Question Stats:

43% (02:35) correct 57% (02:20) wrong based on 317 sessions

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What is the greatest prime factor of $$42^2+75^2+6300$$

a) 3
b) 7
c) 11
d) 13
e) 79
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Joined: 16 Oct 2010
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Location: Pune, India
Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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12 Mar 2015, 21:24
5
8
mvictor wrote:
is there a faster way than squaring all the numbers?

42^2 = (50-8)(50-8) = 1764
75^2 = (100-25)(100-25) = 5625

1764+5625+6300 = 13689
this number is divisible by 3 and 9

divide by 9 and get 1521, the sum of this number is again a multiple of 9, thus we can divide it one more time by 9. we get 169, and 169 is 13^2. thus the answer is 13.

First, take out whatever common number you can find:

$$42^2+75^2+6300 = 3^2 * (14^2 + 25^2 + 700)$$

Now, you may actually know these squares so calculations are a little easier. Of course, it will be great if you can identify that 2*14*25 = 700

$$= 3^2 * ( 14 + 25)^2 = 3^2 * 39^2 = 3^2 * 3^2 * 13^2$$

So 13 is the greatest prime factor.
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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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12 Mar 2015, 12:27
is there a faster way than squaring all the numbers?

42^2 = (50-8)(50-8) = 1764
75^2 = (100-25)(100-25) = 5625

1764+5625+6300 = 13689
this number is divisible by 3 and 9

divide by 9 and get 1521, the sum of this number is again a multiple of 9, thus we can divide it one more time by 9. we get 169, and 169 is 13^2. thus the answer is 13.
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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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12 Mar 2015, 12:55
Hint

A^2+ B^2+2*AB
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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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13 Mar 2015, 06:30
ynaikavde wrote:
What is the greatest prime factor of 42^2+75^2+6300

a) 3
b) 7
c) 11
d) 13
e) 79

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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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04 Jun 2016, 04:19
ynaikavde wrote:
Hint

A^2+ B^2+2*AB

Thanks !!

6300 = 2 * 42* 75

So 42^2 + 75^2 + 6300 = (42+75)^2 = 117^2 = (9*13)^2

So 13
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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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22 Nov 2016, 01:45
1
Great Question.
There are Two ways to do this =>
First Method
Recognise 2*42*75=6300
hence 42^2+75^2+6300= (42+75)^2= 117^2 => 3^4 *13^2 => Greatest Prime factor =13
Second method

Go Traditional way. Open the brackets and get to work.
Lets see

N=42^2+75^2+6300
N=(2*3*7)^2 + (5^2*3)^2 + 2^2*3^2*5^2*7
N=2^2*3^2*7^2 + 5^4*3^2 +2^2*3^2*5^2*7
N=3^2[4*49+625+700]
N=3^2*1521
N=3^2*3^2*169
N=3^4*13^2
Greatest Prime factor is 13

Personally i like the second method.

Hence D
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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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29 Nov 2016, 11:53
42^2 + 75^2 + 6300 = 13689

13689 --> 3x4563 ....169 = 13^2

Thus, the correct answer is D.
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Re: What is the greatest prime factor of 422+752+6300  [#permalink]

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27 Jan 2019, 22:40
ynaikavde wrote:
What is the greatest prime factor of $$42^2+75^2+6300$$

a) 3
b) 7
c) 11
d) 13
e) 79

Always remember to simplify

$$42^2+75^2+6300$$

$$2^2 3^2 7^2 + 3^2 5^4 + 7 * 3^2 2^2 5^2$$

take $$3^2$$ common out

$$3^2 ( 7^2 2^2 + 5^2 [ 5^2 + 28])$$

$$3^2 ( 196 + 25 * 53)$$

$$3^2 ( 1521)$$

3^2 3^2 13^2

Highest will be 13

D
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Re: What is the greatest prime factor of 422+752+6300   [#permalink] 27 Jan 2019, 22:40
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