What is the greatest number less than 1000 which when divided respectively by 5, 7 and 9 leaves the remainders 3, 5 and 7 respectively ?
943
963
953
989
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CISF-AC-EXE – 2018
N ≡ 3 (mod 5)
N ≡ 5 (mod 7)
N ≡ 7 (mod 9)
Notice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9.
Thus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9.
Since 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315.
So, N + 2 = 315k for some integer k.
N = 315k – 2.
We are looking for the greatest number less than 1000.
For k=1, N = 315(1) – 2 = 313.
For k=2, N = 315(2) – 2 = 630 – 2 = 628.
For k=3, N = 315(3) – 2 = 945 – 2 = 943.
For k=4, N = 315(4) – 2 = 1260 – 2 = 1258, which is greater than 1000.
The greatest number less than 1000 is 943.