# Factorials and Exponent Challenge

• MHB
• anemone
In summary, a factorial is a mathematical operation that multiplies a positive integer by all smaller positive integers. To calculate a factorial, you can use a calculator or a formula. An exponent is a notation that indicates the number of times a number is multiplied by itself. To solve an exponent challenge, you can use exponent rules or a calculator. The relationship between factorials and exponents is that they both involve multiplication and can be seen as a base raised to a power.
anemone
Gold Member
MHB
POTW Director
Find all positive integer solutions $(a,\,b,\,c,\,n)$ of the equation $2^n=a!+b!+c!$.

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

Last edited:
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
above one has mistake in last 3 lines

it should be

a = 4 gives 32 power of 2 so n = 5 so solution (4,3,2,5)
a = 5 gives 128 power of 2 so n = 7 so solution (5,3,2,7)

so solution set $(2,1,1,2), (3,1,1,3), (4,3,2,5),(5,3,2,7)$ and any permutation of 1st 3 numbers is each set

## What is a factorial?

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.

## What is an exponent?

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, 34 (read as "3 to the power of 4") is equal to 3 x 3 x 3 x 3 = 81.

## What is the difference between a factorial and an exponent?

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.

## What is the purpose of the "Factorials and Exponent Challenge"?

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.

## How can I improve my skills in working with factorials and exponents?

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.

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