MHB Find A,B,C for Factorial Equation

[solakis1]

find A,B,C such that:

ABC= A!+B!+C!

 

[solakis1]

hint
[sp] since 7!= 7x6x5x4x3x2x1=5040 none of the A,B,C CAN be 7,8,9 because 5040 is a 4 digit no and we are looking for a 3 digit no[/sp]

 

[kaliprasad]

Spoiler: My Solution

Because this is a 3 digit number each digit shall be < 7 as 7! is a 4 digit number that is 5040
Now no digit can be 6 because 6! = 720 so abc >700 so aother digit becomes 7
Largest digit has be 5 because if largest digit is 4 or more we have $ABC \le 72( 3 * 4!)$
So one digit is 5
Now 5!= 120
2 digits cannot be 5 because 5! + 5! = 240 and one dgit has to be 2 and it does not satisfy.
So we have 1 digit 5 and anothee digit A = 1.
B cannot be 5 because then largest C is 4 and sum <= 145 but b =5 makes abc > 150
So C is 5 and we have 100 + 10 * B + 5 = 1 + B! + 120 $ but tying different values B = 4 so number 145

 

[solakis1]

Spoiler: My Solution

Because this is a 3 digit number each digit shall be < 7 as 7! is a 4 digit number that is 5040
Now no digit can be 6 because 6! = 720 so abc >700 so aother digit becomes 7
Largest digit has be 5 because if largest digit is 4 or more we have $ABC \le 72( 3 * 4!)$
So one digit is 5
Now 5!= 120
2 digits cannot be 5 because 5! + 5! = 240 and one dgit has to be 2 and it does not satisfy.
So we have 1 digit 5 and anothee digit A = 1.
B cannot be 5 because then largest C is 4 and sum <= 145 but b =5 makes abc > 150
So C is 5 and we have 100 + 10 * B + 5 = 1 + B! + 120 $ but tying different values B = 4 so number 145

very good but can you explicity mention which are the theorem ,definitions or axiom used to solve the above?

 

[kaliprasad]

very good but can you explicity mention which are the theorem ,definitions or axiom used to solve the above?

Spoiler

I have used the properties of factorial and positional value of numbers . other than that if you want what explicit prpoperties I have used it is nothing.