Euclid’s Division Lemma

Step 1: Substitute in (a=bq+r): (82=6b+4). Step 2: (82-4=78), so (6b=78) and (b=13). Step 3: To find the unknown divisor, subtract the remainder first.

Step 1: (25 \times 4=100). Step 2: (101-100=1), so the quotient is (4) and the remainder is (1). Step 3: The remainder should be neither negative nor equal to the divisor.

Step 1: The Euclidean division form is (a=bq+r). Step 2: Here the divisor is (5) and the remainder is (4), so the form is (5q+4). Step 3: In a general form, multiply the divisor by (q) and add the remainder.

Step 1: The range of remainder starts from (0). Step 2: If the number is exactly divisible by (8), the remainder is (0). Step 3: The smallest possible remainder is always (0).

Step 1: In (a=bq+r), the final added part is the remainder. Step 2: In (76=9 \times 8+4), (4) is less than (9), so it is the remainder. Step 3: Identify the terms by observing the form.

Step 1: Compare with the form (a=bq+r). Step 2: The number multiplied with (9) is (8), so the quotient is (8). Step 3: The quotient is written as the multiplier of the divisor.

Step 1: The remainder cannot be (6) because it equals the divisor. Step 2: (6q+6=6(q+1)+0), so the correct remainder is (0). Step 3: When the remainder seems equal to the divisor, increase the quotient by one.

Step 1: On division by (7), remainders can be from (0) to (6). Step 2: There are (7) possible remainders in total. Step 3: For a divisor (b), the number of possible remainders is (b).

Step 1: (n=2q) means (n) is exactly divisible by (2). Step 2: A number divisible by (2) is an even number. Step 3: Remember the form (2q) for identifying even numbers.

Step 1: In (2q+1), division by (2) leaves remainder (1). Step 2: Such a number is an odd number. Step 3: Every odd number can be written in the form (2q+1).

Step 1: (16 \times 9=144). Step 2: (150-144=6), so (150=16 \times 9+6). Step 3: The remainder (6) is less than (16), so the form is correct.

Step 1: (15) cannot be the remainder because it is greater than (10). Step 2: (15=10+5), so (10q+15=10(q+1)+5). Step 3: The correct remainder is always less than the divisor.

Step 1: When (r=0), the form becomes (a=bq). Step 2: This means (a) is exactly divisible by (b). Step 3: Zero remainder is a sign of exact divisibility.

Step 1: (17 \times 12=204). Step 2: (17 \times 13=221), which is greater than (208), so the quotient is (12). Step 3: While choosing the quotient, also check the next multiple.

Step 1: (17 \times 12=204). Step 2: (208-204=4), so the remainder is (4). Step 3: Check that the remainder (4<17).

Step 1: The remainder must always be less than (9). Step 2: The greatest integer less than (9) is (8). Step 3: The greatest remainder is (b-1).

Step 1: Dividing a number by itself gives quotient (1). Step 2: Nothing remains, so (35=35 \times 1+0). Step 3: When the dividend and divisor are equal, the remainder is (0).

Step 1: Write (n=5q+3). Step 2: (n+1=5q+4), so the new remainder is (4). Step 3: For a small increase, first add it to the old remainder.

Step 1: Write (n=5q+4). Step 2: (n+1=5q+5=5(q+1)+0). Step 3: When the old remainder is one less than the divisor and (1) is added, the new remainder becomes (0).

Step 1: (6 \times 10=60). Step 2: (64-60=4), so (64=6 \times 10+4) is correct. Step 3: The remainder (4) is less than the divisor (6).