Euclid’s Division Lemma

Correct answer: A. The new number will be exactly divisible by (7)

Explanation: Step 1: The original number is (7q+6). Step 2: Adding (1) gives (7q+7=7(q+1)). Step 3: Therefore, the new number is exactly divisible by (7).

Correct answer: A. (0)

Explanation: Step 1: The number is of the form (9q+8). Step 2: Adding (1) gives (9q+9=9(q+1)). Step 3: Therefore, the new remainder is (0).

Correct answer: A. (q=8,\ r=3)

Explanation: Step 1: (4 \times 8=32) and (4 \times 9=36). Step 2: (36) is greater than (35), so (q=8). Step 3: (35-32=3), so (r=3).

Correct answer: A. The remainder is less than the divisor

Explanation: Step 1: The lemma gives the remainder range (0 \le r < b). Step 2: This means the remainder is always less than the divisor. Step 3: In statement-based questions, this rule is very useful.

Correct answer: A. (a) is exactly divisible by (b)

Explanation: Step 1: (r=0) means nothing is left after division. Step 2: Then (a=bq), so (a) is exactly divisible by (b). Step 3: Use zero remainder to identify divisibility.

Correct answer: A. (1)

Explanation: Step 1: (14 \times 7=98). Step 2: (99-98=1), so the remainder is (1). Step 3: The nearest smaller multiple helps find the remainder quickly.

Correct answer: A. (7)

Explanation: Step 1: (14 \times 7=98) and (14 \times 8=112). Step 2: (112) is greater than (99), so the quotient is (7). Step 3: For the quotient, choose a multiple less than or equal to the dividend.

Correct answer: A. (29)

Explanation: Step 1: For remainder (5), the number should be of the form (12q+5). Step 2: (29=12 \times 2+5), so its remainder is (5). Step 3: Check each option by division.

Correct answer: A. (45)

Explanation: Step 1: To leave remainder (3) on division by (7), the form should be (7q+3). Step 2: (45=7 \times 6+3). Step 3: The remainder is less than the divisor, so (45) is correct.

Correct answer: A. (0)

Explanation: Step 1: A number exactly divisible by (10) is of the form (10q). Step 2: Such a number has units digit (0). Step 3: In division by (10), use the units digit for a quick decision.

Correct answer: A. The quotient is (6) and the remainder is (6)

Explanation: Step 1: Compare with (a=bq+r). Step 2: In (84=13 \times 6+6), (q=6) and (r=6). Step 3: Remainder (6) is less than divisor (13), so the form is correct.

Correct answer: A. (16)

Explanation: Step 1: On division by (16), remainders can be from (0) to (15). Step 2: These are (16) values in total. Step 3: For a divisor (b), there are (b) possible remainders.

Correct answer: A. (a<b)

Explanation: Step 1: Putting (q=0) gives (a=b \times 0+r), so (a=r). Step 2: Since (r<b) is necessary, (a<b). Step 3: When the dividend is smaller than the divisor, the quotient can be (0).

Correct answer: A. (5=9 \times 0+5)

Explanation: Step 1: The dividend (5) is smaller than the divisor (9). Step 2: So the quotient is (0) and the remainder is (5). Step 3: Note that remainder (5) is less than (9).

Correct answer: A. Exactly divisible by (3)

Explanation: Step 1: The original number is (3q+2). Step 2: Adding (1) gives (3q+3=3(q+1)). Step 3: Therefore, the new number is exactly divisible by (3).

Correct answer: A. (1)

Explanation: Step 1: (10 \times 12=120). Step 2: (121-120=1), so the remainder is (1). Step 3: In division by (10), the units digit helps find the remainder.

Correct answer: A. (12)

Explanation: Step 1: (10 \times 12=120) and (10 \times 13=130). Step 2: (130) is greater than (121), so the quotient is (12). Step 3: For the quotient, always take the nearest smaller multiple.

Correct answer: A. (70=9 \times 7+7)

Explanation: Step 1: The correct remainder must be at least (0) and less than (9). Step 2: In (70=9 \times 7+7), the remainder is (7), which is less than (9). Step 3: Negative or too large remainders are not accepted.

Correct answer: A. (64=6 \times 10+4)

Explanation: Step 1: (6 \times 10=60) and (6 \times 11=66). Step 2: (60) is the correct smaller multiple and (64-60=4). Step 3: Since (4<6), (64=6 \times 10+4) is the correct form.

Correct answer: A. (0)

Explanation: Step 1: The number is of the form (11q+10). Step 2: Adding (1) gives (11q+11=11(q+1)). Step 3: Therefore, the new remainder is (0).

Correct answer: A. The remainder must be less than the divisor

Explanation: Step 1: The remainder range is (0 \le r < b). Step 2: Here (r=21) and (b=21), so the condition (r<b) is not satisfied. Step 3: A remainder is never written equal to the divisor.

Correct answer: A. Quotient (9), remainder (6)

Explanation: Step 1: Compare with the form (a=bq+r). Step 2: (16) is the divisor, (9) is the quotient, and (6) is the remainder. Step 3: Remainder (6) is less than (16), so the form is correct.

Correct answer: A. (0)

Explanation: Step 1: The original number is (5q+1). Step 2: Adding (4) gives (5q+5=5(q+1)). Step 3: The new number is exactly divisible by (5), so the remainder is (0).

Correct answer: A. (73=12 \times 6+1)

Explanation: Step 1: (12 \times 6=72) and (12 \times 7=84). Step 2: (72) is the correct smaller multiple and (73-72=1). Step 3: Remainder (1) is less than (12), so the form is correct.

Correct answer: A. (0)

Explanation: Step 1: The remainder condition is (0 \le r < b). Step 2: When (b=1), we get (0 \le r < 1), so only (r=0) is possible. Step 3: Any integer divided by (1) leaves remainder (0).