Aspire Faculty ID #15052 · Topic: JAMIA MILLIA ISLAMIA MCA 2018 · Just now
JAMIA MILLIA ISLAMIA MCA 2018

Find the least number which leaves a remainder of 3 when divided by 5, 6, 7 and 8, but leaves no remainder when divided by 9.

Solution

Let $N\equiv 3\pmod{\operatorname{lcm}(5,6,7,8)}$. $\operatorname{lcm}(5,6,7,8)=840$. So $N=840k+3$. For divisibility by $9$: $840k+3\equiv 0\pmod{9} \Rightarrow 3k+3\equiv 0\pmod{9} \Rightarrow k\equiv 2\pmod{3}$. Smallest $k=2 \Rightarrow N=1680+3=1683$.
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