SOLUTION: i am a three digit number less than 200. i am divisible by 9 and 12 .my units digit is less than my ten digit .what am I

Question 171304: i am a three digit number less than 200. i am divisible by 9 and 12 .my units digit is less than my ten digit .what am I
Answer by Mathtut(3670) (Show Source): You can put this solution on YOUR website!
to be divisible by 9 the digits added together must be divisible by 9 and to be divisible by 12 it must be even.
along with units digit being < tens digit....let see if we can come up with something.
ones unit must be 0,2,4,6,or 8.
if 0 then tens unit can be 1-9 case1
if 2 then tens unit can be 3-9 case2
if 4 then tens unit can be 5-9 case3
if 6 then tens unit can be 7-9 case4
if 8 then tens unit can be 9 case5
:
108 is covered in case1 because 0<8
:
lets work backwards
:
case5: 98----9+8=17 to be divisible by 9 hunderds unit has to be 1 and 198
is not divisible by 12
:
case4:96,86,76...96---9+6=16 to be div by 9 100's unit has to be 3 and that is more than 200. 86 ...8+6=14 to be div by 9 100's unit has to be 4....to big
same problem with 76
:
case3:94,84,74,64,54...9+4, 8+4,7+4,6+4,5+4 all present problems with #'s larger than 200
:
case2:92,82,72,62,52,42,32 ...92,82,and 72 all get throw out because #'s larger than 200. 62 is 6+2=8..to be divisble by 9 100's unit must be 1...162..not divisible by 12 take 52,42,32 end up with being over 200
:
case1:90,80,70,60,50,40,30,20,10... 90 larger than 200...80 is 8+0=8 in order to be divisible by 9 100's unit has to be 1. 180.......guess what this number is divisible by 9 and 12 and is less than 200
applying divisibility rules for the rest of the numbers in this case make all the numbers to large >200
:
SO WE HAVE GONE THROUGH ALL THE CASES AND 180 is the only one that qualified
there could be a simpler way but I will have to think on that
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