Euclid’s Division Lemma

Correct answer: C. Quotient

Explanation: Step 1: In (a=bq+r), (q) is the quotient. Step 2: The number multiplied with (47) is (13), so (13) is the quotient. Step 3: To identify the quotient, look at the multiplier written with the divisor.

Correct answer: D. Remainder

Explanation: Step 1: In (a=bq+r), the final added part is (r). Step 2: Here (7) is less than (47), so it is the remainder. Step 3: While identifying the remainder, also check its range.

Correct answer: C. (17)

Explanation: Step 1: Substitute in (a=bq+r): (93=5b+8). Step 2: (85=5b), so (b=17). Step 3: To find the unknown divisor, subtract the remainder first.

Correct answer: C. (7)

Explanation: Step 1: Substitute in (a=bq+r): (108=15q+3). Step 2: (105=15q), so (q=7). Step 3: To find the unknown quotient, subtract the remainder first.

Correct answer: A. (14q+9)

Explanation: Step 1: The Euclidean division form is (a=bq+r). Step 2: Here the divisor is (14) and the remainder is (9), so the form is (14q+9). Step 3: In a general form, multiply the divisor by (q) and add the remainder.

Correct answer: C. (9q+9)

Explanation: Step 1: On division by (9), remainders can be from (0) to (8). Step 2: In (9q+9), the remainder is (9), which equals the divisor. Step 3: It should be written correctly as (9(q+1)).

Correct answer: B. (10)

Explanation: Step 1: (33 \times 30=990). Step 2: (1000-990=10), so the remainder is (10). Step 3: For a large dividend, finding a nearby multiple saves time.

Correct answer: B. (30)

Explanation: Step 1: (33 \times 30=990) and (33 \times 31=1023). Step 2: Since (1023) is greater, the quotient is (30). Step 3: The quotient is the greatest integer whose product with the divisor does not exceed the dividend.

Correct answer: A. (3)

Explanation: Step 1: (2a=2(5q+4)=10q+8). Step 2: (8=5+3), so (2a=5(2q+1)+3) and the remainder is (3). Step 3: In multiplication-based questions, first multiply the old remainder and then divide by the divisor.

Correct answer: A. (1)

Explanation: Step 1: The remainder of (a) is (5), so for (3a), check (3 \times 5=15). Step 2: (15=7 \times 2+1), so the remainder is (1). Step 3: In such questions, work with the remainder instead of the whole number.

Correct answer: C. (2)

Explanation: Step 1: Write (a=4q+3). Step 2: (2a=8q+6=4(2q+1)+2), so the remainder is (2). Step 3: After multiplication, do not forget to convert the remainder into the correct range.

Correct answer: B. (4)

Explanation: Step 1: Let (a=6q+5). Step 2: (2a=12q+10=6(2q+1)+4), so the remainder is (4). Step 3: (10) cannot be the remainder because it is greater than (6).

Correct answer: A. (24)

Explanation: Step 1: The remainder is always less than the divisor. Step 2: The greatest integer less than (25) is (24). Step 3: The greatest remainder can be found quickly using (b-1).

Correct answer: A. (0)

Explanation: Step 1: A remainder may start from (0). Step 2: If a number is exactly divisible by (25), the remainder is (0). Step 3: The smallest possible remainder is always (0).

Correct answer: A. (39=100 \times 0+39)

Explanation: Step 1: (39) is smaller than (100), so the quotient is (0). Step 2: (39=100 \times 0+39), and (39<100), so the form is correct. Step 3: If the dividend is smaller, the remainder can be the dividend itself.

Correct answer: B. (q=2, r=22)

Explanation: Step 1: (39 \times 2=78) and (39 \times 3=117). Step 2: Since (117) is greater, (q=2) and (r=100-78=22). Step 3: The remainder must be less than (39).

Correct answer: A. (0)

Explanation: Step 1: Write (a=bq+(b-1)). Step 2: (a+1=bq+b=b(q+1)+0), so the remainder is (0). Step 3: Adding (1) to the greatest remainder brings the remainder back to (0).

Correct answer: B. (1)

Explanation: Step 1: (a=bq) is exactly divisible by (b). Step 2: (a+1=bq+1), so the remainder is (1). Step 3: Just after an exactly divisible number, the remainder is (1).

Correct answer: C. (23)

Explanation: Step 1: (28 \times 9=252) and (28 \times 10=280). Step 2: Since (280) is greater, the remainder is (275-252=23). Step 3: If the next multiple is greater, choose the previous multiple.

Correct answer: B. (9)

Explanation: Step 1: (28 \times 9=252) and (28 \times 10=280). Step 2: Since (280) is greater than (275), the quotient is (9). Step 3: Choose the quotient whose product does not exceed the dividend.

Correct answer: A. (0)

Explanation: Step 1: Adding (14) to the old remainder (12) gives (26). Step 2: (26=13 \times 2+0), so the new remainder is (0). Step 3: Do not forget to divide the new sum by the same divisor.

Correct answer: A. (11)

Explanation: Step 1: (a-3=13q-2), but the remainder should not be negative. Step 2: (13q-2=13(q-1)+11), so the remainder is (11). Step 3: When a negative remainder appears, add the divisor to make the correct remainder.

Correct answer: A. (0)

Explanation: Step 1: In (a=42q+41), the remainder is (41), one less than (42). Step 2: (a+1=42q+42=42(q+1)+0), so the remainder is (0). Step 3: Adding (1) to a (b-1) remainder gives exact division.

Correct answer: A. (3)

Explanation: Step 1: In (a=42q+40), the remainder is (40). Step 2: (a+5=42q+45=42(q+1)+3), so the remainder is (3). Step 3: If the sum crosses the divisor, subtract the divisor once.

Correct answer: C. (a=31q+30)

Explanation: Step 1: On division by (31), the remainder must be from (0) to (30). Step 2: In (31q+30), the remainder is (30), which is less than (31). Step 3: A remainder (31), (37), or negative is not in standard form.