Question #139859
The sum of the first two digits of a three digit number is 4. The sum of the numbers
that is formed by using each digit of the number once is 1998. If the number has 8
factors, then what is the number?
1
Expert's answer
2020-10-25T18:34:02-0400

Let the number is abc.\overline{abc}. Then

a+b=4,a0a+b=4, a\not=0


100a+10b+c+100a+b+10c+100a+10b+c+100a+b+10c+

+100b+10a+c+100b+a+10c++100b+10a+c+100b+a+10c+

+100c+10a+b+100c+a+10b=1998+100c+10a+b+100c+a+10b=1998

222(a+b+c)=1998222(a+b+c)=1998

a+b+c=9a+b+c=9

c=94=5c=9-4=5

The possible numbers are


135,225,315,405135, 225, 315, 405

The 8 factors of 135 are: 1,3,5,9,15,27,45,1351,3,5,9,15,27,45,135

The 9 factors of 225 are: 1,3,5,9,15,25,45,75,2251,3,5,9,15,25,45, 75,225

The 12 factors of 315 are: 1,3,5,7,9,15,21,35,45,63,105,3151,3,5, 7,9,15,21, 35, 45, 63,105, 315

The 10 factors of 405 are: 1,3,5,9,15,27,45,81,1305,4051,3,5, 9,15,27, 45, 81,1305, 405


Therefore the number is 135.135.



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