Euclid’s Division Lemma

Step 1: Use (a=bq+r) for the dividend. Step 2: (a=15 \times 7+4=105+4=109). Step 3: First match the given words with symbols, then calculate.

Step 1: (41) is smaller than (60), so the quotient is (0). Step 2: (41=60 \times 0+41), and (41<60), so the form is correct. Step 3: When the dividend is smaller, the remainder can be the dividend itself.

Step 1: The remainder is smaller than the divisor. Step 2: The greatest integer less than (19) is (18). Step 3: When the greatest remainder is asked, use (b-1).

Step 1: The range of the remainder starts from (0). Step 2: If a number is exactly divisible by (19), the remainder is (0). Step 3: The smallest possible remainder is always (0).

Step 1: Write (n=7q+6). Step 2: (n+1=7q+7=7(q+1)+0), so the new remainder is (0). Step 3: If the old remainder is one less than the divisor and (1) is added, the new remainder becomes (0).

Step 1: Write (n=8q+5). Step 2: (n+6=8q+11=8(q+1)+3), so the remainder is (3). Step 3: If the new sum exceeds the divisor, subtract the divisor from it.

Step 1: Let (n=9q+4). Step 2: (n+3=9q+7), so the remainder on division by (9) is (7). Step 3: Adding the increase to the old remainder is a quick method.

Step 1: (27 \times 11=297) and (27 \times 12=324). Step 2: Since (324) is greater than (310), the quotient is (11). Step 3: Always check the next multiple when choosing the quotient.

Step 1: (27 \times 11=297). Step 2: (310-297=13), so the remainder is (13). Step 3: The final remainder must be less than the divisor (27).

Step 1: The remainder cannot be (12) because it equals the divisor. Step 2: (12q+12=12(q+1)+0), so the correct remainder is (0). Step 3: If the remainder equals the divisor, increase the quotient by one.

Step 1: (46) is (2) times (23). Step 2: (23q+46=23(q+2)+0), so the remainder is (0). Step 3: If the added part is a multiple of the divisor, the remainder can become (0).

Step 1: In the Euclidean form (a=bq+r), (b) is the divisor and (r) is the remainder. Step 2: Here the divisor is (5) and the remainder should be (2), so the form is (5q+2). Step 3: In a general form, multiply the divisor by (q) and add the remainder.

Step 1: On division by (12), remainders can be from (0) to (11). Step 2: Their total number is (12). Step 3: For a divisor (b), the number of possible remainders is (b).

Step 1: (31 \times 12=372) and (31 \times 13=403). Step 2: Since (403) is greater, (400=31 \times 12+28). Step 3: A form with a negative remainder is not the standard Euclidean form.

Step 1: (a=6q+5). Step 2: (a+1=6q+6=6(q+1)+0), so the remainder is (0). Step 3: Adding (1) to a remainder that is one less than the divisor gives exact division.

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.

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.

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.

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).

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).