- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Find all positive integer solutions $(a,\,b,\,c,\,n)$ of the equation $2^n=a!+b!+c!$.
kaliprasad said:Without loss of generalty we can have $a>=b>=c$.
Now we get a multiple of power of 2 only when we add multiples of same power of 2
So $a!$ and $b!+c!$ should be muiltiple of same power of 2. and when we add the 2
we shall get multiple of power of 2 say of the form $m2^x$. if m is power of 2 then we are
done.
Now b and c should be multiple of same power of 2 and when we add the same we get a multiple of
higher power and further this should be same as multiple of power of 2 of a.
So we have 2 cases to check
$b = c$ and $a = 2$ or 3 (for reason please see below )-
And $ b = c + 1$ then any a.
As c devides a!+b!+c! so c can not be greater than 2 as sum is power of 2
Now
Put the values b= 1, c = 1 giving a = 2 or 3 as a =4 gives a! divsible by 4 but b!+c! is not
a =2 gives c = 2
a =3 gives c = 3
c= 2, b= 3 gives b! + c! = 8 so we need to check for a = 4 and 5 only as a = 6 or above a! is
divisible by 16 so it is not possible
a = 4 gives 32 power of 2 so n = 5 so solution (4,3,2,7)
a = 5 gives 128 power of 2 so n = 7 so solution (5,3,2,7)
so solution set $(2,1,1,2), (3,1,1,3), (4,3,2,7),(4,3,2,7)$ and any permutation of 1st 3 numbers is each set
A factorial is a mathematical function represented by an exclamation mark (!) that is used to calculate the product of a given number and all the positive integers below it. For example, 5! (read as "5 factorial") is equal to 5 x 4 x 3 x 2 x 1 = 120.
An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself. It is denoted by a superscript number placed to the right of the base number. For example, 3^{4} (read as "3 to the power of 4") is equal to 3 x 3 x 3 x 3 = 81.
A factorial is used to calculate the product of a given number and all the positive integers below it, while an exponent is used to calculate the result of a base number being multiplied by itself a certain number of times. In other words, a factorial is a specific type of exponent where the base number is a positive integer and the exponent is also a positive integer.
The "Factorials and Exponent Challenge" is a problem-solving activity that aims to improve one's understanding and skills in working with factorials and exponents. It involves solving a series of mathematical equations involving factorials and exponents, and finding the correct answers in the shortest amount of time.
One way to improve your skills in working with factorials and exponents is to practice solving different types of problems involving these mathematical concepts. You can also read and study mathematical concepts related to factorials and exponents, and seek help from a math tutor or teacher if needed.