find common factor of 43 raised to power 43+47 raised to power 43&43 raised to power 47+47 raised to power 47?

Answer 1

See

http://answers.yahoo.com/question/index;_ylt=AtlXqHmCmv.FU4B3VNmqJ6UjzKIX?qid=20061207214522AA32Fgo

Answer 2

Suppose p is a divisor both of 43^43 +47^43 and of 43^47+47^47, then it will be a divisor of
(43^4+47^4)(43^43 +47^43) - ( 43^47+47^47),

which is (47^4)( 43^43) +(43^4)(47^43) which can also be written as

43^4 * 47^4 * (47^39+43^39). But since 43 and 47 are prime p is relatively prime with 43 and 47. Therefore p must divide 47^39+43^39. Doing the same reasoning we see that p must divide 47^3 + 43 ^3 which is 183330=2*3*3*3*5*17*61

Now if p divides 47^3 + 43 ^3 and 43^7 + 47^7 it divides

(43^4+47^4)(43^3 +47^3) - ( 43^7+47^7) which is 47^3*43^3(47+43). Therefore, it must divide 90. Since 90 divides
183330, this is the answer.

Answer 3

I assume you were asking about finding the least common factor of 43^90^43 and 43^94^47. Well...to put things in perspective, the least common factor of only 2^3^4 and 3^4^5 is 513 digits. I could not compute this with a 32-bit computer. Maybe someone with a supercomputer will help? Good luck!

Answer 4

the question is as i think
43^43 + 47^43 and 43^47 + 47^47

then common factor will be simply 2*43^43

Answer 5

The question to which the link above points is not necessarily the same as that asked here. That other question uses parentheses to make it VERY clear that the "+" sign refers to the addition of two numbers, each of them a straightforward power of 43.

Perhaps mistakenly, I gave the questioner credit for having removed the parentheses for a specific purpose. I presumed that that purpose included not wanting to have people simply pasting in a solution from the other site, but to ask a possibly subtly different question.

Unfortunately, the usual rules for resolving mathematical ambiguities don't help us completely, here. (Recall the acronym PEMDAS or "Please excuse my dear Aunt Sally.") The dilemma boils down to "If math and language are mixed, do we resolve any language issues first, or last?"

I confess that I don't know if there can be a hard and fast rule here. I realise now that I first did the only mathematical operations specified (without even a gap between the operation sign and each one of the pair of numbers, which I took to mean "really add these first"), then turned to interpret the language. I now see that others did it differently, interpreting the language first, then applying math rules to that. I find it intriguing but a little bit depressing that these CAN lead to different interpretations. The problem is related to the general one of expressing a mathematical question partly in words, rather than clearly and properly written mathematics.

I think we can all agree that the aim should be, however, to set a clear, not a possibly ambiguous question --- as well as not merely re-setting one already resolved. To that end, I appeal (as I have done, over and over again) for questioners to do everything they can to express their questions in mathematical terms, to anticipate ambiguities, and to use parentheses etc. to clear them up in advance.

So, for what it's worth (perhaps: not much!) here was my own original interpretation:

The answer is the first number, [43^(43 + 47)]^43, or the incredibly large 43 ^ 3870 (!)

This is because it divides exactly into the LARGER number,
[43^(47 + 47)^47].

(Note that you asked for "common factor," of which there are an ENORMOUS NUMBER! Normally, "common factor" is used as part of a unique specification, the "highest common factor," or HCF of two numbers. This is defined as: "the largest positive integer that divides both numbers without remainder." ***See below.*** One responder has confused this with "least common factor," which, for any two integers is plain old 1." If one number divides into the other without remainder, it is necessarily the HCF.)

One can show the complete division property by working out the exponent from doing the division. It's:

(47 + 47) 47 - (43 + 47) 43 = (47^2 - 43^2) + 47(47 - 43)

= (47 - 43) (47 + 43 + 47) = 548.

So the result of doing this division is 43^ 548. (This is itself an extremely large number that can't be shown here, yet it is dwarfed by even the smaller of the two numbers you gave us to examine in the question.)

Live long and prosper.

Answer 6

the question is as i think
43^43 + 47^43 and 43^47 + 47^47

then common factor will be simply 2*43^43

for more clarification you may ask directly at http://cbsemaths.co.in online leaving your e-mail address

Answer 7

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Answer 8

I THINK 1