Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| [a, b as given in the Euclidean algorithm, quotient (q), Remainder (r)] \(a=bq+r\) | Remainder (r) | ||
| (A) | \(a=112,\ b=7\) | (I) | \(r=1\) |
| (B) | \(a=118,\ b=9\) | (II) | \(r=3\) |
| (C) | \(a=119,\ b=6\) | (III) | \(r=5\) |
| (D) | \(a=115,\ b=8\) | (IV) | \(r=0\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
Euclid’s division lemma writes any number \(a\) as \(a=bq+r\), where \(q\) is the quotient and the remainder \(r\) satisfies \(0\le r\lt b\). So \(r\) is found by taking the largest multiple of \(b\) that does not exceed \(a\) away from \(a\).
(A) \(a=112,\ b=7\) — this matches (IV) \(r=0\).
\(7\times16=112\) uses up all of 112, so nothing is left: \(112=7\times16+0\)
(B) \(a=118,\ b=9\) — this matches (I) \(r=1\).
The largest multiple of 9 up to 118 is \(9\times13=117\), leaving 1: \(118=9\times13+1\)
(C) \(a=119,\ b=6\) — this matches (III) \(r=5\).
The largest multiple of 6 up to 119 is \(6\times19=114\), leaving 5: \(119=6\times19+5\)
(D) \(a=115,\ b=8\) — this matches (II) \(r=3\).
The largest multiple of 8 up to 115 is \(8\times14=112\), leaving 3: \(115=8\times14+3\)