Question

Given that a, b, and c are positive integers, solve the following equation.
a!b! = a! + b! + 2^c

Answer 1

no guessing!

Answer 2

the answer gives small numbers, so you can guess it

Answer 3

but that is not fair :-)

Answer 4

wait I want to solve it algebraically

Answer 5

good good, hint: start by dividing with b! or a!

Answer 6

assume that a>=b (without the loss of generality) now we divide by b!

a!=a!/b!+1+2^c/b!
As each term of the RHS is an integer RHS>=3

this was the second hint :-)

Answer 7

Well, the good thing is that the only time 2^c/b! can be an integer is if b! is a power of 2. The only time b! is a power of 2 is when b!=2, and so b=2. Now that b!=2, we can see that a!=a!/2+1+2^c/2. We can see that a!-a!/2=a!/2, so a!/2=1+2^c/2. Take out the 1/2 and we're left with a!=2+2^c.

Answer 8

nearly but b=2 or b=1

Answer 9

? I got b=2

Answer 10

for b=1it does not make sense as a!=a!+1+2^c
so b=2

Answer 11

a!/2 = 1 + 2^(c-1)

Answer 12

OHHOHO I see

Answer 13

is a is greater than 3 a!/2 is even that cannot be so

Answer 14

a!/2-1 has to be even

Answer 15

so a has to be 3

Answer 16

right nice

Answer 17

c=2

Answer 18

yup a=3 b=2 c=2

Answer 19

I liked this one too :-)

Answer 20

This one was challenging but it was possible (with hints :)

Answer 21

these are 1 level more complex, should I look something from tier 4? (hardest)

Answer 22

lol go for it

Answer 23

I hardly understand the questions :)

Answer 24

The divisors of a positive integer, excluding the number itself, are called the proper
divisors . If the sum of proper divisors is equal to the number we call the number
perfect. For example, the divisors of 28 are 1, 2, 4, 7, 14, and 28, so the sum of
proper divisors is 1 + 2 + 4 + 7 + 14 = 28.
The first eight perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056,
137438691328, 2305843008139952128.
Prove that P is an even perfect number iff it is of the form 2^(n-1) * (2^n -1) where
2^n -1 is prime.

Answer 25

these are sick :-) I have no clue how to do this