My teacher asked me the following question:
If $a \equiv 8 \mod{16}$ then $2a \equiv 16 \mod{32}$.
I solved this problem by plugging in a bunch of numbers but I wonder if I can prove it in a more abstract way?
Thanks in advance.
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That' s obvious from the formula of the Euclidean division. Furthermore, you also have $2a\equiv 0\mod 16$. – Bernard Apr 03 '20 at 21:46
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1Use the definition. What does it mean that $a\equiv8\bmod16$? Then, what can you say about $2a$? – Andrea Mori Apr 03 '20 at 21:46
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1Maybe try the definition of modular congruence, i mean, try to prove that, if $16\vert (a-8)$ then $32\vert(2a - 16)$ – Ulivai Apr 03 '20 at 21:46
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1$a\equiv8\pmod{16}$ means that $(a-8)/16$ is an integer. $2a\equiv16\pmod{32}$ means that $(2a-16)/32$ is an integer. – Angina Seng Apr 03 '20 at 22:06
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For the reverse inference - cancelling or division - see here. – Bill Dubuque Apr 03 '20 at 22:24