Euclid’s Division Lemma

Step 1: The remainder of (N) is 11. Step 2: Adding 25 gives total remainder (11+25=36), and 36 is exactly divisible by 18. Step 3: After addition, divide the total remainder again by the divisor.

Step 1: When the divisor is 19, the remainder can be from 0 to 18. Step 2: 19 is equal to the divisor, so it cannot be a remainder. Step 3: In remainder-range questions, watch carefully for the option equal to the divisor.

Step 1: (53\times27=1431). Step 2: (1441-1431=10), so the remainder is 10. Step 3: While dividing, choose the multiple that does not exceed the number.

Step 1: The remainder of (t) is 9. Step 2: The remainder part of (2t+3) is (2\times9+3=21), and (21=16+5). Step 3: Do not forget to reduce the final remainder below 16.

Step 1: Let the number be (20q+3). Step 2: For five times the number, the remainder is (5\times3=15), which is less than 20. Step 3: In multiplication, multiply the remainder and check the limit.

Step 1: Among two consecutive integers, one is even, so a factor 2 is present. Step 2: Among three consecutive integers, one is divisible by 3, so a factor 3 is present. Step 3: Since 2 and 3 together make 6, the product is divisible by 6.

Step 1: A number of the form (4q+1) is odd. Step 2: The square of an odd number leaves remainder 1 when divided by 8; for example, (1^2) and (5^2) both leave remainder 1. Step 3: Remember the rule that odd squares leave remainder 1 on division by 8.

Step 1: (a-35=29q+20-35=29q-15). Step 2: This can be written as (29(q-1)+14), so the remainder is 14. Step 3: Add the divisor to a negative remainder to make it valid.

Step 1: For the square, take (8^2=64). Step 2: (64=13\times4+12), so the remainder is 12. Step 3: Do not write 64 as the final answer because a remainder must be smaller than the divisor.

Step 1: The remainder of (x) is 5. Step 2: The remainder of (x^2+x) comes from (5^2+5=30), and 30 is divisible by 6. Step 3: Substitute the remainder in the expression to solve quickly.

Step 1: (17\times20=340) and (17\times21=357). Step 2: The nearest lower multiple of 17 below 352 is 340, so the remainder is 12. Step 3: Do not accept a negative remainder or a remainder greater than 17.

Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (32\times15+31=480+31=511). Step 3: Remainder 31 is less than divisor 32, so the form is valid.

Step 1: For the difference, the remainder is (9-10=-1). Step 2: Add 14 to make it a valid remainder, giving 13. Step 3: In subtraction, add the divisor when the remainder becomes negative.

Step 1: Any integer divided by 4 is of the form (4q), (4q+1), (4q+2), or (4q+3). Step 2: Four consecutive integers cover all these four remainders. Step 3: The one with remainder 0 is divisible by 4.

Step 1: The original remainder is 26. Step 2: Adding 28 gives total remainder (26+28=54), which is exactly divisible by 27. Step 3: It is easier to reduce the added number by the divisor and combine remainders.

Step 1: The remainder of (u) is 7. Step 2: The remainder of (u^2-u) comes from (7^2-7=42), and (42=10\times4+2). Step 3: Substitute the remainder first, then find the final remainder.

Step 1: For the fourth power, look at the remainder of (4^4). Step 2: (4^2=16), which leaves remainder 2 when divided by 7; then (4^4) leaves the same remainder as (2^2=4). Step 3: For higher powers, reduce remainders step by step.

Step 1: The number is (15q+10). Step 2: Adding 5 gives (15q+15=15(q+1)), so the remainder is 0. Step 3: When the remainder and added number together make the divisor, the new remainder becomes 0.

Step 1: The remainders of (a) and (b) are 8 and 5. Step 2: The remainder of (ab+a) comes from (8\times5+8=48), and 48 is divisible by 12. Step 3: In a mixed expression, find the remainder of each part separately.

Step 1: In standard form, the remainder is not negative. Step 2: (100\times9=900), so (999=900+99), and 99 is less than 100. Step 3: A form with a negative remainder may look computationally close, but it is not Euclidean form.