the number is factorials that digit number
Algebra
Level
2
abc is a digit number and it equal with a! + b! +c!. What is abc???
n! = 1x2x3x4x...xn
197
475
145
719
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1 solution
6!=720, 7!=5040 so a,b,c<7 If one of them is 6 then abc>722 (abc is a digit number) so, a>6 which is contradiction. So a,b,c<6 a=b=c=5 is not solution so a<3(because abc<120+120+24), For a=2 ,candidate for b and c is only number 5. But 255 is not equal to 2+120+120=242,so a<2 For a=1 we get that one of b or c must be 5,other must be less than 5. If b=5 then 150+c=1+120+c! and from that 29+c=c! which is not true. If we assume that c=5 then we get that 105+10b=1+b!+120 That means 10b-b!=16,and b=4 is solution. So a=1,b=4,c=5 is our solution,i.e. number 145 is answer.