Problem No. 44
Number Theory
Level
4
How many three digit positive integers are there, the sum of whose digits is a perfect cube?
The answer is 38.
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1 solution
Greatest three digit integer is 9 9 9 = 3 3 . So only we can have 1^3=1,~~~2^3=8,~~~3^3=27.\\ 1^3=~1=1+0+0,~that~ is~ 100~~ONLY.\\ 3^3=27=9+9+9,~that ~is~ 999~~ONLY.\\ 2^3=~8~has~many~options.\\ Left~ digit~~1,.~middle~can~be~0~to~7,~~~8~~options, right ~no~option. \\ Left~ digit~~2,.~middle~can~be~0~to~6,~~~7~~options, right ~no~option. \\ ' ' ' ' ' ' '\\ ' ' ' ' ' ' '\\ Left~ digit~~8,.~middle~can~be~0~Only,~~~1~~options, right ~no~option. \\ So ~total=~1+1+8+7+6+5+4+3+2+1=2+\dfrac{8*9} 2=~~~\Large \color{#D61F06}{38}