Euclid’s Division Lemma

Step 1: Compare with (a=bq+r). Step 2: In (a=4q+2), (r=2). Step 3: Since (2<4), this remainder is valid.

Step 1: Euclid’s division lemma is applied to two positive integers. Step 2: (b) is the divisor and it cannot be zero. Step 3: For the dividing number, positivity and non-zero value are necessary.

Step 1: (14 \times 8=112). Step 2: (119-112=7), so the remainder is (7). Step 3: Do not forget to check that the final remainder is less than the divisor.

Step 1: (9) cannot be the remainder because it is greater than (7). Step 2: (9=7+2), so (7q+9=7(q+1)+2). Step 3: A large remainder must be converted into the correct range.

Step 1: The main condition on the remainder is (0 \le r < b). Step 2: This means the remainder is less than the divisor. Step 3: In theory-based questions, remember this condition directly.

Step 1: In the form (a=bq+r), (b) is the divisor. Step 2: Here (13) is multiplied by (6), so (13) is the divisor. Step 3: While identifying terms, look at the first number in the product.

Step 1: The Euclidean form is (a=bq+r). Step 2: Here (b=10) and (r=6), so the number is (10q+6). Step 3: Put the divisor and the remainder in the correct places.

Step 1: The dividend (49) is smaller than the divisor (50). Step 2: The quotient is (0) and the remainder remains (49). Step 3: When a smaller number is divided by a larger number, the remainder can be the smaller number itself.

Step 1: The remainder is less than the divisor. Step 2: The greatest integer less than (18) is (17). Step 3: If (b) is given, the greatest remainder is (b-1).

Step 1: Write (n=8q+3). Step 2: (n+4=8q+7), so the remainder is (7). Step 3: Add the added number to the old remainder and check whether the sum is less than the divisor.

Step 1: (11 \times 12=132) and (11 \times 13=143). Step 2: (132) is the nearest smaller multiple of (11), so the remainder is (134-132=2). Step 3: The remainder in the answer must always be less than the divisor.

Step 1: In Euclid’s division lemma, the remainder is not negative. Step 2: The remainder is smaller than the divisor, so (0 \le r < 14) is correct. Step 3: Taking (r=14) would be wrong because the remainder cannot equal the divisor.

Step 1: A remainder can start from (0). Step 2: When dividing by (10), the remainder must be less than (10), so values from (0) to (9) are possible. Step 3: Do not include the divisor among possible remainders.

Step 1: Use the Euclidean form (a=bq+r). Step 2: (a=12 \times 9+5=108+5=113). Step 3: To find the number, first multiply the divisor and quotient, then add the remainder.

Step 1: On division by (6), the possible remainders are (0,1,2,3,4,5). Step 2: So the number is written as (6q+r) using these remainders. Step 3: While forming general forms, list all possible remainders in order.

Step 1: (13 \times 6=78) and (87-78=9). Step 2: Since (9) is less than (13), (87=13 \times 6+9) is correct. Step 3: A negative remainder or a remainder greater than the divisor does not give the correct Euclidean form.

Step 1: (18 \times 12=216). Step 2: (221-216=5), so the remainder is (5). Step 3: Confirm that the remainder (5<18).

Step 1: The remainder must be less than (16), but (21) is larger. Step 2: (21=16+5), so (16q+21=16(q+1)+5). Step 3: If a large remainder appears, divide it again by the divisor and correct it.

Step 1: In (a=bq+r), (a) is the number being divided. Step 2: Such a number is called the dividend. Step 3: Remembering the meanings of symbols helps solve questions quickly.

Step 1: Substitute in (a=bq+r): (97=8b+1). Step 2: (96=8b), so (b=12). Step 3: To find the unknown divisor, subtract the remainder first.