Euclid’s Division Lemma

Correct answer: C. (5)

Explanation: Step 1: In (a=6q+1), the old remainder is (1). Step 2: (a+4=6q+5), so the new remainder is (5). Step 3: If the new remainder remains less than the divisor, it is the answer.

Correct answer: C. (9)

Explanation: Step 1: Compare with (a=bq+r). Step 2: The number multiplied with (8) is (9), so (9) is the quotient. Step 3: To identify the quotient, look at the multiplier written with the divisor.

Correct answer: B. (8)

Explanation: Step 1: In the Euclidean form, (b) is the divisor. Step 2: In (75=8 \times 9+3), (8) is the number by which division is done. Step 3: Identify the divisor by looking at the first number in the product.

Correct answer: B. (q=1, r=0)

Explanation: Step 1: A number divided by itself gives quotient (1). Step 2: (28=28 \times 1+0), so (q=1) and (r=0). Step 3: When dividend and divisor are the same, remember that the remainder is (0).

Correct answer: B. Odd number

Explanation: Step 1: Remainder (1) on division by (2) means the number has the form (2q+1). Step 2: A number of the form (2q+1) is odd. Step 3: To identify an odd number, focus on remainder (1).

Correct answer: B. Even number

Explanation: Step 1: Remainder (0) means the number is exactly divisible by (2). Step 2: A number exactly divisible by (2) is even. Step 3: The general form of an even number is (2q).

Correct answer: A. (3q+3=3(q+1))

Explanation: Step 1: (a=3q+2) is given. Step 2: (a+1=3q+3=3(q+1)), so it is exactly divisible by (3). Step 3: While changing the form, add the added (1) to the remainder.

Correct answer: C. (38)

Explanation: Step 1: (40 \times 12=480) and (40 \times 13=520). Step 2: Since (520) is greater, the remainder is (518-480=38). Step 3: Avoid the next larger multiple and always use the nearest smaller multiple.

Correct answer: C. (1)

Explanation: Step 1: (30) cannot be the remainder because it is greater than (29). Step 2: (29q+30=29(q+1)+1), so the correct remainder is (1). Step 3: Always keep the remainder between (0) and (b-1).

Correct answer: C. (28)

Explanation: Step 1: Match (a=29q+28) with (a=bq+r). Step 2: Here (r=28), and (28<29), so it is a valid remainder. Step 3: If the remainder is less than the divisor, the form is already correct.

Correct answer: A. Every positive integer (a) can be written as (bq+r) where (0 \le r < b)

Explanation: Step 1: The main form of Euclid’s division lemma is (a=bq+r). Step 2: The remainder is greater than or equal to (0) and less than the divisor. Step 3: In theory questions, remembering the range of the remainder is important.

Correct answer: A. Dividend (45) is smaller than divisor (72)

Explanation: Step 1: (0 \times 72+45=45), so the dividend is (45). Step 2: The divisor is (72), and (45<72), so the quotient is (0) and the remainder is (45). Step 3: When the quotient is (0), the dividend is usually smaller than the divisor.

Correct answer: A. (0)

Explanation: Step 1: In (a=20q+19), the remainder is (19). Step 2: (a+1=20q+20=20(q+1)+0), so the remainder is (0). Step 3: Adding (1) to a remainder (b-1) makes the new remainder (0).

Correct answer: A. (3)

Explanation: Step 1: (a=20q+18). Step 2: (a+5=20q+23=20(q+1)+3), so the remainder is (3). Step 3: If the sum exceeds (20), subtract (20) to get the new remainder.

Correct answer: C. (27)

Explanation: Step 1: (37 \times 27=999). Step 2: So the quotient is (27) and the remainder is (0). Step 3: If the product is exactly equal to the dividend, that multiplier is the quotient and the remainder is (0).

Correct answer: A. (0)

Explanation: Step 1: (37 \times 27=999). Step 2: (999-999=0), so the remainder is (0). Step 3: In exact division, the remainder is always (0).

Correct answer: A. (13q+8)

Explanation: Step 1: The Euclidean form is (a=bq+r). Step 2: Here the divisor is (13) and the remainder is (8), so the form is (13q+8). Step 3: In a general form, place the divisor with (q) and the remainder at the end.

Correct answer: D. (3q+3)

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

Correct answer: A. (0)

Explanation: Step 1: Write the number as (4q+3). Step 2: The next number is (4q+4=4(q+1)+0). Step 3: After the greatest remainder, the next number has remainder (0) again.

Correct answer: B. (1)

Explanation: Step 1: The number has the form (5q). Step 2: The next number is (5q+1), so the remainder is (1). Step 3: In consecutive numbers, remainders increase in order and then return to (0).

Correct answer: B. (9)

Explanation: Step 1: (9 \times 9=81) and (9 \times 10=90). Step 2: Since (90) is greater than (85), the quotient is (9). Step 3: For the quotient, take the greatest multiplier whose product does not exceed the dividend.

Correct answer: B. (4)

Explanation: Step 1: Taking (q=9), we get (9 \times 9=81). Step 2: (85-81=4), so (r=4). Step 3: After finding the remainder, check that (4<9).

Correct answer: A. (a) is a multiple of (b)

Explanation: Step 1: When (r=0), the equation becomes (a=bq). Step 2: This means (a) is a multiple of (b). Step 3: Remember zero remainder as a sign of exact divisibility.

Correct answer: C. (32)

Explanation: Step 1: (50 \times 13=650). Step 2: (682-650=32), so the remainder is (32). Step 3: In division by (50), the remainder must be less than (50).

Correct answer: A. (1)

Explanation: Step 1: In (a=50q+49), the remainder is (49). Step 2: (a+2=50q+51=50(q+1)+1), so the remainder is (1). Step 3: If the new remainder exceeds the divisor, subtract the divisor from it.