Euclid’s Division Lemma

Correct answer: A. Quotient (7), remainder (8)

Explanation: Step 1: Dividing (71) by (9), we get (9 \times 7=63). Step 2: (71-63=8), so the quotient is (7) and the remainder is (8). Step 3: Always check that the remainder is smaller than the divisor.

Correct answer: A. (0 \le r < 12)

Explanation: Step 1: In Euclid’s division lemma, the remainder is always greater than or equal to (0) and smaller than the divisor. Step 2: Here the divisor is (12), so (0 \le r < 12). Step 3: Writing (\le 12) is wrong because the remainder cannot equal the divisor.

Correct answer: B. (q=8, r=1)

Explanation: Step 1: (13 \times 8=104), the closest smaller multiple of (13) to (105). Step 2: (105-104=1), so (q=8) and (r=1). Step 3: In such questions, the remainder must not be negative or greater than the divisor.

Correct answer: B. (0,1,2,3,4)

Explanation: Step 1: When a number is divided by (5), the remainder must be smaller than (5). Step 2: The remainder may also be (0), so the possible remainders are (0,1,2,3,4). Step 3: Remember that a remainder is never equal to the divisor.

Correct answer: C. (81)

Explanation: Step 1: Use the Euclidean form (a=bq+r). Step 2: (a=7 \times 11+4=77+4=81). Step 3: In such questions, multiply the divisor and quotient first, then add the remainder.

Correct answer: B. (96=15 \times 6+6)

Explanation: Step 1: The remainder must be smaller than (15). Step 2: (15 \times 6=90) and (96-90=6), so the correct form is (96=15 \times 6+6). Step 3: Check not only equality but also the condition on the remainder.

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

Explanation: Step 1: On division by (3), the possible remainders are (0,1,2). Step 2: Therefore, the number can be written as (3q+0, 3q+1, 3q+2). Step 3: Build general forms using possible remainders.

Correct answer: B. Incorrect, because the remainder must be less than (8)

Explanation: Step 1: In Euclid’s division lemma, the remainder satisfies (0 \le r < b). Step 2: Here (b=8), so (r) cannot be equal to (8). Step 3: Remembering the range of the remainder is very important.

Correct answer: A. (1)

Explanation: Step 1: Divide (43) by (6). Step 2: (6 \times 7=42) and (43-42=1), so (r=1). Step 3: While choosing the quotient, make sure (bq) does not exceed (a).

Correct answer: B. (7)

Explanation: Step 1: Check multiples of (17). Step 2: (17 \times 7=119) and (17 \times 8=136), which is greater than (126). So the quotient is (7). Step 3: The quotient is the greatest integer for which the product does not exceed the number.

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

Explanation: Step 1: On division by (4), the possible remainders are (0,1,2,3). Step 2: Hence the forms are (4q,4q+1,4q+2,4q+3). Step 3: These forms are very useful in questions related to even and odd numbers.

Correct answer: B. (q=8, r=6)

Explanation: Step 1: (19 \times 8=152) and (19 \times 9=171). Step 2: (152) is less than (158), so (q=8) and (r=158-152=6). Step 3: If the next multiple exceeds the number, take the previous multiple.

Correct answer: B. Divisor

Explanation: Step 1: In the form (a=bq+r), (a) is the dividend. Step 2: (b) is the number by which division is done, so it is the divisor. Step 3: Keep the meaning of symbols clear to avoid calculation errors.

Correct answer: A. (89=10 \times 8+9)

Explanation: Step 1: The remainder should not be negative and must be less than (10). Step 2: (10 \times 8=80) and the remainder is (9), so (89=10 \times 8+9). Step 3: A form with a negative remainder is not the standard Euclidean form.

Correct answer: C. (52)

Explanation: Step 1: Substitute the values in (a=bq+r). Step 2: (a=9 \times 5+7=45+7=52). Step 3: You can check the result by dividing it again by (9).

Correct answer: B. (8)

Explanation: Step 1: Find the nearest multiple of (24) less than (200). Step 2: (24 \times 8=192) and (200-192=8), so the remainder is (8). Step 3: Practice finding nearby multiples for faster calculation.

Correct answer: A. (13)

Explanation: Step 1: Compare with the Euclidean form (a=bq+r). Step 2: Here the divisor is (17) and the remainder is (13), since (13<17). Step 3: Always check the range of the remainder while comparing.

Correct answer: B. (a=11(q+1)+4)

Explanation: Step 1: The remainder must be smaller than (11). Step 2: (15=11+4), so (11q+15=11(q+1)+4). Step 3: If the remainder is greater than the divisor, divide it again and rewrite the form.

Correct answer: A. (6q)

Explanation: Step 1: Remainder (0) means the number is exactly divisible by (6). Step 2: So the number is (6q+0), that is (6q). Step 3: Forms with zero remainder help identify multiples.

Correct answer: B. (257=32 \times 8+1)

Explanation: Step 1: (32 \times 8=256). Step 2: (257-256=1), so (257=32 \times 8+1). Step 3: In the correct form, the remainder must be smaller than (32) and not negative.

Correct answer: A. Two positive integers

Explanation: Step 1: This lemma is used for two positive integers. Step 2: In it, (a) and (b) are positive integers and (b \ne 0). Step 3: While reading the question, pay attention to the type of numbers involved.

Correct answer: B. Quotient (12), remainder (5)

Explanation: Step 1: (30 \times 12=360). Step 2: (365-360=5), so the quotient is (12) and the remainder is (5). Step 3: Keeping the remainder smaller than the divisor decides whether the answer is valid.

Correct answer: C. (4)

Explanation: Step 1: Compare it with (a=bq+r). Step 2: In (a=5q+4), the divisor is (5) and the remainder is (4). Step 3: Since (4<5), this form is already correct.

Correct answer: A. (2q) and (2q+1)

Explanation: Step 1: On division by (2), the remainder can only be (0) or (1). Step 2: So the forms are (2q) or (2q+1). Step 3: This is the basis for identifying even and odd numbers.

Correct answer: A. (8)

Explanation: Step 1: (20 \times 7=140). Step 2: (148-140=8), so (r=8). Step 3: If an option gives a remainder equal to or greater than (20), reject it immediately.