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What is the remainder when 7^74-5^74 is divided by 24? Is

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What is the remainder when 7^74-5^74 is divided by 24? Is [#permalink] New post 03 Jul 2008, 12:19
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What is the remainder when 7^74-5^74 is divided by 24?

Is there an easy way to solve this?
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Re: PS: What is the easiest way to solve? [#permalink] New post 03 Jul 2008, 13:20
wizardofwashington wrote:
What is the remainder when 7^74-5^74 is divided by 24?

Is there an easy way to solve this?


easiest way for me: 7^74 - 5^74 = (49)^37-25^37 = (24*2+1)^37 - (24+1)^37 -> remainder is 1^37 - 1^37 = 0
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Re: PS: What is the easiest way to solve? [#permalink] New post 03 Jul 2008, 13:45
In addition to maraticus's solution:

It may be useful to remember that a^n-b^n is always divisible by (a-b).

So, when we write 49^37-25^37, we can note that 49-25=24, and thus, the expression can be evenly divided by 24.
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Re: PS: What is the easiest way to solve? [#permalink] New post 03 Jul 2008, 13:57
greenoak wrote:
In addition to maraticus's solution:

It may be useful to remember that a^n-b^n is always divisible by (a-b).

So, when we write 49^37-25^37, we can note that 49-25=24, and thus, the expression can be evenly divided by 24.


Thanks, guys. Appreciate your quick response.
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Re: PS: What is the easiest way to solve? [#permalink] New post 04 Jul 2008, 21:58
greenoak wrote:
In addition to maraticus's solution:

It may be useful to remember that a^n-b^n is always divisible by (a-b).

So, when we write 49^37-25^37, we can note that 49-25=24, and thus, the expression can be evenly divided by 24.


Ditto :)
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Re: PS: What is the easiest way to solve? [#permalink] New post 04 Jul 2008, 22:20
maratikus wrote:
wizardofwashington wrote:
What is the remainder when 7^74-5^74 is divided by 24?

Is there an easy way to solve this?


easiest way for me: 7^74 - 5^74 = (49)^37-25^37 = (24*2+1)^37 - (24+1)^37 -> remainder is 1^37 - 1^37 = 0


I like your approach. however i figured out as under:

7^1 - 5^1 = 2 so reminder = 2.
7^2 - 5^2 = 24 ...... so reminder 0.
7^3 - 5^3 = 48 ...... so reminder 0.
7^4 - 5^4 = 218 ....... so reminder = 2

similarly;
7^73 - 5^73 should have a reminder of 2.
7^74 - 5^74 should have a reminder of 0.
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Re: PS: What is the easiest way to solve?   [#permalink] 04 Jul 2008, 22:20
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