please. . .anyone, is there a quick way of expressing 275840 as a product of prime powers!? i have one way but it'll take aaaaages! thanks
the number is even, so it is divisible by two. it is also divisible by 5. so start dividing it by prime numbers as much as possible. eventually you will see the number as a product of primes. if a prime repeats itself in the product, just rewrite it as that prime raised to the appropriate power.
example, lets do what i said for 360
divide by 2 we get: 180
divide by 2 again we get: 90
divide by 2 again, we get: 45
two can't go into this, what other prime i know can? let's try 5
divide by 5 we get: 9
aha, now divide by 3 to get 3 and again by 3 to get 1, and we're done.
so, 360 = 2 x 2 x 2 x 5 x 3 x 3 = $\displaystyle 2^3 \cdot 3^2 \cdot 5$
Long took me three minutes at most...
$\displaystyle 275840 = 10*27584$
$\displaystyle =(2*5)*(4*6896)$
$\displaystyle =(2^3*5)*(6896)$
$\displaystyle =(2^3*5)*(6900-4)$
$\displaystyle =(2^3*5)*(4*25*69-4)$
$\displaystyle =(2^5*5)*(25*69-1)$
$\displaystyle =(2^5*5)*(1724)$
$\displaystyle =(2^5*5)*(4*431)$
$\displaystyle =(2^7*5)*(431)$
You sais you wanted 275840 as an expression of two prime powers ... well, it is divisible by 10, since it ends with a zero. And 10 is diisible by 5 and 2. Now, since they are factors in 10 they must be in 275840 as well! 5 and 2 is primes, and you said that you wanted exactly two prime powers. So, powers of 2 and 5 it is then. You can start with finding the exponent over any of the two numbers, I start with 5:
1: 275840 / 5 = 55168 which is not divisible by 5 any more. The exponent over 5 is 1.
1: 55168 / 2 = 27584
2: 27584 / 2 = 13792
3: 13792 / 2 = 6896
4: 6896 / 2 = 3448
5: 3448 / 2 = 1724
6: 1724 / 2 = 862
7: 862 / 2 = 431 which is not divisible by 2 any more. The exponent over 2 is 7, but you will still need to have more prime powers before you can write 275840 as a product of them. And remember that a number can only be expressed as a product of prime powers in one way, so it can't be expressed as a product of two prime powers. Sorry kid, it can't be done.