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111A09201B/9 has a remainder of 5. find the least value of A+B This is from a competitive mathematics contest and I have no idea how to go about solving it.

If anyone here would like to help me not only help with this question but help improve my skills in this competition which consist of “Basic mathematics, algebra 1, geometry, algebra 2, trigonometry, analysis, basic calculus, probability, and Mis.” Please comment that if you would like to help me. I’m sure that you’ll enjoy helping teach me. Please help mathematicians, I too wish to learn the ways of math.

Tylor Gonzalez
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  • If $x$ is an integer with digits $d_1,\ldots, d_k$, then $$x\equiv d_1+\cdots+d_k\pmod 9$$ – Dave Jan 23 '19 at 03:04
  • I don’t follow could elaborate further? Sorry I’m not the brightest I’ve only taken up to calculus II. – Tylor Gonzalez Jan 23 '19 at 03:04
  • This follows from writing $$x=10^{k-1}d_k+\cdots+10d_2+d_1$$ and recognizing that $10\equiv 1\pmod 9$. – Dave Jan 23 '19 at 03:05

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$$1+1+1+A+0+9+2+0+1+B\equiv5(mod9),$$ which gives $$A+B\equiv8(mod9)$$ and we see that $8$ is the answer.

Michael Rozenberg
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