Euclid’s Division Lemma

Correct answer: A. (a=bq+r,\ 0\le r<b)

Explanation: Step 1: The lemma connects dividend, divisor, quotient, and remainder. Step 2: The correct form is (a=bq+r), where the remainder is at least (0) and less than the divisor. Step 3: Always check the range of the remainder in exams.

Correct answer: A. Quotient (5), remainder (5)

Explanation: Step 1: We can write (35=6\times5+5). Step 2: The quotient is (5) and the remainder is (5), which is less than (6). Step 3: The remainder must never be equal to or greater than the divisor.

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

Explanation: Step 1: The range of the remainder is very important in the lemma. Step 2: The remainder may be (0), but it must be less than the divisor (b). Step 3: Read the inequality carefully in such questions.

Correct answer: A. (q=5,\ r=2)

Explanation: Step 1: The greatest multiple of (9) not exceeding (47) is (45). Step 2: So (47=9\times5+2), giving (q=5) and (r=2). Step 3: First find the nearest smaller multiple of the divisor.

Correct answer: A. (q=7,\ r=3)

Explanation: Step 1: (7\times7=49) and (52-49=3). Step 2: Since (3) is less than (7), (q=7,\ r=3) is correct. Step 3: Reject negative remainders or remainders equal to the divisor.

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

Explanation: Step 1: Remainders start from (0) and go up to one less than the divisor. Step 2: The divisor is (8), so possible remainders are (0) to (7). Step 3: Do not include the divisor itself as a possible remainder.

Correct answer: A. (3)

Explanation: Step 1: In (5k+3), the part (5k) is a multiple of (5). Step 2: A multiple of (5) leaves remainder (0), so the final remainder is (3). Step 3: Learn to identify (r) in the form (bq+r).

Correct answer: A. Every positive integer can be written as a multiple of the divisor plus a remainder

Explanation: Step 1: The lemma gives a systematic way to express division. Step 2: In (a=bq+r), (bq) is a multiple of the divisor and (r) is the remainder. Step 3: In meaning-based questions, understand both the formula and the words.

Correct answer: A. (1)

Explanation: Step 1: In (a=bq+r), the small added part is the remainder. Step 2: In the given expression, (1) is less than (9), so it is the remainder. Step 3: Do not interchange quotient and remainder.

Correct answer: A. (0)

Explanation: Step 1: (13\times7=91). Step 2: The number is exactly divisible, so the remainder is (0). Step 3: In exact division, remember to write the remainder as (0).

Correct answer: A. (29=4\times7+1)

Explanation: Step 1: (4\times7=28), the nearest smaller multiple of (4) to (29). Step 2: (29-28=1), so the correct form is (29=4\times7+1). Step 3: Do not allow the remainder to be negative or equal to (4).

Correct answer: A. (10)

Explanation: Step 1: The remainder is always less than the divisor. Step 2: The greatest integer less than (11) is (10). Step 3: The greatest possible remainder is always one less than the divisor.

Correct answer: A. The dividend is exactly divisible by the divisor

Explanation: Step 1: Remainder (0) means nothing is left after division. Step 2: Therefore, the dividend is exactly divisible by the divisor. Step 3: To identify exact divisibility, check the remainder.

Correct answer: A. (76=10\times7+6)

Explanation: Step 1: (10\times7=70) and (76-70=6). Step 2: The remainder (6) is less than (10), so the form is correct. Step 3: When dividing by (10), the last digit often helps find the remainder.

Correct answer: A. (0,1,2,3,4,5)

Explanation: Step 1: Here the divisor is (6). Step 2: The remainder can be from (0) to (5), because it must be less than (6). Step 3: Include (0) in the list of possible remainders.

Correct answer: A. (3)

Explanation: Step 1: (5\times3=15). Step 2: (18-15=3), so the remainder is (3). Step 3: Choose a multiple that does not exceed the dividend.

Correct answer: A. Quotient

Explanation: Step 1: In (a=bq+r), (a) is the dividend and (b) is the divisor. Step 2: (q) represents the quotient and (r) represents the remainder. Step 3: Remembering the meaning of symbols helps solve short questions quickly.

Correct answer: A. Remainder

Explanation: Step 1: The lemma is written as (a=bq+r). Step 2: Here (r) is the part left after division, so it is the remainder. Step 3: Identify (q) and (r) separately.

Correct answer: A. (4)

Explanation: Step 1: (9\times4=36). Step 2: (40-36=4), so the remainder is (4). Step 3: In mental calculation, first take the nearest smaller multiple.

Correct answer: A. (12)

Explanation: Step 1: In (a=bq+r), (b) is the divisor. Step 2: In the given form, (12) is multiplied by (q), so it is the divisor. Step 3: The number multiplying the quotient is usually the divisor.

Correct answer: A. (4)

Explanation: Step 1: In (a=bq+r), (q) is the quotient. Step 2: In (12\times4), the number (4) is in the place of the quotient. Step 3: Match the whole form to identify divisor and quotient.

Correct answer: A. (23=5\times3+8)

Explanation: Step 1: To find the invalid form, check the range of the remainder. Step 2: In (23=5\times3+8), the remainder (8) is greater than the divisor (5). Step 3: Even if the equality is numerically true, check the remainder condition.

Correct answer: A. (3)

Explanation: Step 1: When the divisor is (3), possible remainders are (0,1,2). Step 2: Remainder (3) equals the divisor, so it is not possible. Step 3: Treat a remainder equal to the divisor as incorrect.

Correct answer: A. (10)

Explanation: Step 1: (10\times10=100). Step 2: (101=10\times10+1), so the quotient is (10). Step 3: The quotient is decided by the nearest smaller multiple.

Correct answer: A. (37=6\times6+1)

Explanation: Step 1: (6\times6=36). Step 2: (37-36=1), and (1<6), so the form is correct. Step 3: Wrong options can be removed quickly by checking the remainder range.