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# What is the smallest prime factor of 5^8+10^6–50^3?

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What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 07:21
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Question Stats:

53% (01:42) correct 47% (01:46) wrong based on 359 sessions

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What is the smallest prime factor of $$5^8+10^6–50^3$$?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for a correct solution.

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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Updated on: 25 Jan 2015, 09:33
3
5 is the smallest prime factor.

5^8+10^6–50^3 = 5^6 * 97

And since 97 is a prime number, 5 is the smallest prime.

EDIT: Man, i'm such an idiot. I missed the minus

5^8+10^6–50^3 = 5^6 * 81

3 is the smallest prime factor

Originally posted by HaseebR7 on 23 Jan 2015, 07:28.
Last edited by HaseebR7 on 25 Jan 2015, 09:33, edited 1 time in total.
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 07:47
2
ans B 3....
5^8+10^6–50^3= 5^6(5^2+2^6-2^3)
=5^6*81....so 3 is the smallest prime factor
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 13:00
5^8 has 5 as a prime factor.

10^6 has 2 and 5 as prime factors.

50^3 has 2 and 5 as prime factors.

Thus only 2 and 5 are possible answers. The answer is 2 if the solution is even and 5 if the solution is odd. Eliminate BDE.

5^8 is odd because products of odd integers are always odd.

10^6 and 50^3 are even because products of even integers are always even.

Odd + Even - Even = Odd.

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 13:15
1
Hi HaseebR7,

You have the right idea, but you should double-check your math.

Hi CCMBA,

You have to very careful when trying to apply "math theory" to complicated situations. If your understanding is off (even a little bit), then you'll get one of the wrong answers and not even know it.

This question is really about re-writing exponents and finding ways to combine 'like' terms. Try using exponent rules against this question...

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 16:17
Okay, so I tried to recombine and this is what I got.

5^8 - 10^6 + 50^3

5^8 - (5^6 * 2^6) + (5^3 * 5^3 *2^3)

5^8 - 5^6(2^3 * 2^3) + (5^6 * 2^3)

5^8 - [5^6 * 2^3](2^3 + 1)

5^6 * 5^2 - {5^6 * 2^3](2^3 + 1)

5^6 [5^2 - (2^3 *2^3 + 1)]

5^6 (25 - 65)

5^6(-40)

-5^6 * 2^2 * 2 *5 --> answer is 2?

I kept getting turned around, but I *think* the math is finally right. I would be guessing on this question. I think recognizing that the answer should be 2 or 5 is enough.

EDIT: Nope, I just cheated by breaking out the calculator and got a multiple of 3. Chetan2u makes it look so easy!
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 21:10
1
CCMBA wrote:
Okay, so I tried to recombine and this is what I got.

5^8 - 10^6 + 50^3

5^8 - (5^6 * 2^6) + (5^3 * 5^3 *2^3)

5^8 - 5^6(2^3 * 2^3) + (5^6 * 2^3)

5^8 - [5^6 * 2^3](2^3 + 1)

5^6 * 5^2 - {5^6 * 2^3](2^3 + 1)

5^6 [5^2 - (2^3 *2^3 + 1)]

5^6 (25 - 65)

5^6(-40)

-5^6 * 2^2 * 2 *5 --> answer is 2?

I kept getting turned around, but I *think* the math is finally right. I would be guessing on this question. I think recognizing that the answer should be 2 or 5 is enough.

EDIT: Nope, I just cheated by breaking out the calculator and got a multiple of 3. Chetan2u makes it look so easy!

There is an error in your concept:

Smallest prime factor of 2*5*11 is 2 - Correct
Smallest prime factor of 2 + 5 is neither 2 nor 5 but actually 7 - a new prime number.
When you add, you don't know the prime factors you will get.

Here is how you will recombine:

$$5^8 + 10^6 - 50^3$$ (You have the negative sign misplaced in your expression)

$$5^8 + 2^6*5^6 - 2^3*5^6$$

$$5^6 * (25 + 64 - 8)$$

$$81 * 5^6$$

$$3^4 * 5^6$$

The smallest prime is 3.
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 21:18
Bunuel wrote:
What is the smallest prime factor of 5^8+10^6–50^3?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for a correct solution.

(5^8)+(10^6)–(50^3)
Breaking each component down gives:
(5^8) smallest prime is 5
(2*5)^6 smallest prime is 2
(2*5^2)^3 smallest prime is 2

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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23 Jan 2015, 21:24
peachfuzz wrote:
Bunuel wrote:
What is the smallest prime factor of 5^8+10^6–50^3?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for a correct solution.

(5^8)+(10^6)–(50^3)
Breaking each component down gives:
(5^8) smallest prime is 5
(2*5)^6 smallest prime is 2
(2*5^2)^3 smallest prime is 2

okay never mind, I need to decompose the equation further so that it turns into (5^6)(81). Hence, 3...
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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24 Jan 2015, 00:54
Smallest prime factor will be 3.

Simplify the equation in the same way as posted by many.

option B it is.
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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26 Jan 2015, 04:17
Bunuel wrote:
What is the smallest prime factor of 5^8+10^6–50^3?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

With exponents questions like this, your guiding principles should include "Find Common Bases" and "Factor" - the composite bases 5, 10, and 50 can all be turned into prime bases of 5s and 2s, and the addition and subtraction can be factored to multiplication. First you should break the 10 and 50 down into prime factors:

5^8+2^6*5^6−2^3*5^6
Then factor the common 5 terms out to create a multiplication problem:

5^6(5^2+2^6−2^3)
Which allows you to deal with the math in the parentheses, since those numbers are all reasonable to calculate by hand:

5^6(2^5+6^4−8)
Which leads to 5^6(81), which factors down to (5^6)(^34). Since those are all prime, and the question asks for the SMALLEST prime, the answer is 3.

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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03 Feb 2015, 01:32
3
$$5^8 + 10^6 – 50^3$$

$$= 5^8 + 5^6 * 2^6 – 5^6 * 2^3$$

$$= 5^6(25+64) - 5^6 * 2^3$$

$$= 5^6(25+64-8)$$

$$= 5^6 * 81$$

$$= 5^6 * 3^4$$

Smallest prime factor = 3
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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08 Feb 2015, 16:22
HaseebR7 wrote:
5 is the smallest prime factor.

5^8+10^6–50^3 = 5^6 * 97

And since 97 is a prime number, 5 is the smallest prime.

EDIT: Man, i'm such an idiot. I missed the minus

5^8+10^6–50^3 = 5^6 * 81

3 is the smallest prime factor

So did I....missed minus.!
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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28 Feb 2015, 03:53
1
chetan2u wrote:
ans B 3....
5^8+10^6–50^3= 5^6(5^2+2^6+2^3)
=5^6*81....so 3 is the smallest prime factor

Shouldn't the part in red above be -2^3 and not +2^3?
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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28 Feb 2015, 04:29
pacifist85 wrote:
chetan2u wrote:
ans B 3....
5^8+10^6–50^3= 5^6(5^2+2^6+2^3)
=5^6*81....so 3 is the smallest prime factor

Shouldn't the part in red above be -2^3 and not +2^3?

hi pacifist,
you are correct...
it was a typing error... edited it thank you
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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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30 Nov 2016, 09:59
5^8 + 10^6 - 50^3
5^8 + (5x2)^6 - (5^2 x 2)^3
5^8 + (5^6)(2^6) - (5^6)(2^3)
5^6(5^2 + 2^6 - 2^3)
5^6(81)

81 --> 3 is lowest prime

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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30 Nov 2016, 11:23
Bunuel wrote:
What is the smallest prime factor of $$5^8+10^6–50^3$$?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for a correct solution.

$$5^8+10^6–50^3$$

= $$5^8+ ( 5^6*2^6 ) – ( 5^6*2^3)$$

= $$5^6 \ [ \ 5^2+ 2^6 – 2^3 \ ]$$

= $$5^6 \ [ \ 5^2+ 2^3 ( 2^3 – 1 ) \ ]$$

= $$5^6 \ [ \ 25+ 8 * 7 \ ]$$

= $$5^6 *81$$

= $$5^6 *3^3$$

Thus , the smallest prime factor will be (B) 3

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Re: What is the smallest prime factor of 5^8+10^6–50^3?  [#permalink]

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15 Jan 2018, 23:06
Bunuel wrote:
What is the smallest prime factor of $$5^8+10^6–50^3$$?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for a correct solution.

$$5^8 + 10^6 - 50^3$$

$$5^8 = (5^2)^4$$

$$10^6 = ((2*5)^2)^3$$

$$50^3 = (2*5^2)^3$$

$$(5^2)^3 + (5^2)^3 - (5^2)^3 [5^2 + 2^6 - 2^3] = 5^6 * 81 = 5^6 * 3^4.$$

So, $$3$$ is the smallest prime factor. Ans - B.
Re: What is the smallest prime factor of 5^8+10^6–50^3?   [#permalink] 15 Jan 2018, 23:06
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