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Jamia Millia Islamia Previous Year Questions (PYQs)

Jamia Millia Islamia Remainder PYQ

Jamia Millia Islamia PYQ
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.





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Jamia Millia Islamia Previous Year PYQ Jamia Millia Islamia JAMIA MILLIA ISLAMIA MCA 2018 PYQ

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 $k=2 \Rightarrow N=1680+3=1683$.
Jamia Millia Islamia PYQ
 A man packs boxes into parcels. If he packs 3, 4, 5 or 6 in a parcel, he is left with one over; if he packs 7 in a parcel, none is left. What is the number of boxes?





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Jamia Millia Islamia Previous Year PYQ Jamia Millia Islamia JAMIA MILLIA ISLAMIA MCA 2021 PYQ

Solution

N ≡ 1 (mod 3,4,5,6) ⇒ N ≡ 1 (mod 60). Also N ≡ 0 (mod 7).
So 60m+1 ≡ 0 (mod 7) ⇒ 4m+1 ≡ 0 ⇒ m ≡ 5 (mod 7) ⇒ smallest N = 60×5+1 = 301.

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