Euclid’s Division Lemma

Step 1: The original number is (6q+2). Step 2: Adding (1) gives (6q+3). Step 3: Since (3<6), the new remainder is (3).

Step 1: (11 \times 4=44) and (11 \times 5=55). Step 2: (55) is greater than (54), so (44) is the correct multiple. Step 3: (54-44=10), so (54=11 \times 4+10) is correct.

Step 1: (17 \times 7=119) and (17 \times 8=136). Step 2: (136) is greater than (120), so the quotient is (7). Step 3: Choose the quotient so that the product does not exceed the dividend.

Step 1: (17 \times 7=119). Step 2: (120-119=1), so the remainder is (1). Step 3: Since (1<17), this remainder is valid.

Step 1: The Euclidean form is (a=bq+r). Step 2: If the divisor is (3) and the remainder is (1), the form is (3q+1). Step 3: The small term at the end of the form shows the remainder.

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).

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).

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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).

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).