Euclid’s Division Lemma

Correct answer: A. Quotient (12), remainder (2)

Explanation: Step 1: (11 \times 12=132) and (11 \times 13=143). Step 2: (132) is the nearest smaller multiple of (11), so the remainder is (134-132=2). Step 3: The remainder in the answer must always be less than the divisor.

Correct answer: A. (0 \le r < 14)

Explanation: Step 1: In Euclid’s division lemma, the remainder is not negative. Step 2: The remainder is smaller than the divisor, so (0 \le r < 14) is correct. Step 3: Taking (r=14) would be wrong because the remainder cannot equal the divisor.

Correct answer: A. (0,1,2,3,4,5,6,7,8,9)

Explanation: Step 1: A remainder can start from (0). Step 2: When dividing by (10), the remainder must be less than (10), so values from (0) to (9) are possible. Step 3: Do not include the divisor among possible remainders.

Correct answer: C. (113)

Explanation: Step 1: Use the Euclidean form (a=bq+r). Step 2: (a=12 \times 9+5=108+5=113). Step 3: To find the number, first multiply the divisor and quotient, then add the remainder.

Correct answer: A. (6q,6q+1,6q+2,6q+3,6q+4,6q+5)

Explanation: Step 1: On division by (6), the possible remainders are (0,1,2,3,4,5). Step 2: So the number is written as (6q+r) using these remainders. Step 3: While forming general forms, list all possible remainders in order.

Correct answer: B. (87=13 \times 6+9)

Explanation: Step 1: (13 \times 6=78) and (87-78=9). Step 2: Since (9) is less than (13), (87=13 \times 6+9) is correct. Step 3: A negative remainder or a remainder greater than the divisor does not give the correct Euclidean form.

Correct answer: B. (5)

Explanation: Step 1: (18 \times 12=216). Step 2: (221-216=5), so the remainder is (5). Step 3: Confirm that the remainder (5<18).

Correct answer: C. (5)

Explanation: Step 1: The remainder must be less than (16), but (21) is larger. Step 2: (21=16+5), so (16q+21=16(q+1)+5). Step 3: If a large remainder appears, divide it again by the divisor and correct it.

Correct answer: A. Dividend

Explanation: Step 1: In (a=bq+r), (a) is the number being divided. Step 2: Such a number is called the dividend. Step 3: Remembering the meanings of symbols helps solve questions quickly.

Correct answer: C. (12)

Explanation: Step 1: Substitute in (a=bq+r): (97=8b+1). Step 2: (96=8b), so (b=12). Step 3: To find the unknown divisor, subtract the remainder first.

Correct answer: C. (109)

Explanation: Step 1: Use (a=bq+r) for the dividend. Step 2: (a=15 \times 7+4=105+4=109). Step 3: First match the given words with symbols, then calculate.

Correct answer: A. (41=60 \times 0+41)

Explanation: Step 1: (41) is smaller than (60), so the quotient is (0). Step 2: (41=60 \times 0+41), and (41<60), so the form is correct. Step 3: When the dividend is smaller, the remainder can be the dividend itself.

Correct answer: A. (18)

Explanation: Step 1: The remainder is smaller than the divisor. Step 2: The greatest integer less than (19) is (18). Step 3: When the greatest remainder is asked, use (b-1).

Correct answer: A. (0)

Explanation: Step 1: The range of the remainder starts from (0). Step 2: If a number is exactly divisible by (19), the remainder is (0). Step 3: The smallest possible remainder is always (0).

Correct answer: A. (0)

Explanation: Step 1: Write (n=7q+6). Step 2: (n+1=7q+7=7(q+1)+0), so the new remainder is (0). Step 3: If the old remainder is one less than the divisor and (1) is added, the new remainder becomes (0).

Correct answer: A. (3)

Explanation: Step 1: Write (n=8q+5). Step 2: (n+6=8q+11=8(q+1)+3), so the remainder is (3). Step 3: If the new sum exceeds the divisor, subtract the divisor from it.

Correct answer: C. (7)

Explanation: Step 1: Let (n=9q+4). Step 2: (n+3=9q+7), so the remainder on division by (9) is (7). Step 3: Adding the increase to the old remainder is a quick method.

Correct answer: C. (11)

Explanation: Step 1: (27 \times 11=297) and (27 \times 12=324). Step 2: Since (324) is greater than (310), the quotient is (11). Step 3: Always check the next multiple when choosing the quotient.

Correct answer: C. (13)

Explanation: Step 1: (27 \times 11=297). Step 2: (310-297=13), so the remainder is (13). Step 3: The final remainder must be less than the divisor (27).

Correct answer: C. (0)

Explanation: Step 1: The remainder cannot be (12) because it equals the divisor. Step 2: (12q+12=12(q+1)+0), so the correct remainder is (0). Step 3: If the remainder equals the divisor, increase the quotient by one.

Correct answer: C. (0)

Explanation: Step 1: (46) is (2) times (23). Step 2: (23q+46=23(q+2)+0), so the remainder is (0). Step 3: If the added part is a multiple of the divisor, the remainder can become (0).

Correct answer: A. (5q+2)

Explanation: Step 1: In the Euclidean form (a=bq+r), (b) is the divisor and (r) is the remainder. Step 2: Here the divisor is (5) and the remainder should be (2), so the form is (5q+2). Step 3: In a general form, multiply the divisor by (q) and add the remainder.

Correct answer: B. (12)

Explanation: Step 1: On division by (12), remainders can be from (0) to (11). Step 2: Their total number is (12). Step 3: For a divisor (b), the number of possible remainders is (b).

Correct answer: A. (400=31 \times 12+28)

Explanation: Step 1: (31 \times 12=372) and (31 \times 13=403). Step 2: Since (403) is greater, (400=31 \times 12+28). Step 3: A form with a negative remainder is not the standard Euclidean form.

Correct answer: A. (0)

Explanation: Step 1: (a=6q+5). Step 2: (a+1=6q+6=6(q+1)+0), so the remainder is (0). Step 3: Adding (1) to a remainder that is one less than the divisor gives exact division.