Euclid’s Division Lemma

Correct answer: A. Because one of them is divisible by 2 and one is divisible by 3

Explanation: 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.

Correct answer: B. 1

Explanation: 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.

Correct answer: A. 14

Explanation: 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.

Correct answer: C. 12

Explanation: 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.

Correct answer: A. 0

Explanation: 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.

Correct answer: A. (q=20, r=12)

Explanation: 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.

Correct answer: A. 511

Explanation: 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.

Correct answer: A. 13

Explanation: 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.

Correct answer: A. Because division by 4 gives the cycle of remainders 0, 1, 2, 3

Explanation: 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.

Correct answer: A. 0

Explanation: 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.

Correct answer: B. 2

Explanation: 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.

Correct answer: D. 4

Explanation: 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.

Correct answer: A. 0

Explanation: 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.

Correct answer: A. 0

Explanation: 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.

Correct answer: B. (999=100\times9+99)

Explanation: 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.

Correct answer: B. 1

Explanation: Step 1: The possible remainders are 1 and 5. Step 2: (1^2=1) and (5^2=25=6\times4+1), so the remainder is 1 in both cases. Step 3: Squares of many numbers coprime to 6 show remainder 1, but checking is always better.

Correct answer: B. 6

Explanation: Step 1: For seven times the number, the remainder part is (7\times4=28). Step 2: (28=22+6), so the final remainder is 6. Step 3: If the result after multiplication is greater than the divisor, reduce it.

Correct answer: A. 0

Explanation: Step 1: The remainder of (a) is 2. Step 2: The remainder of (a^2+1) comes from (2^2+1=5), which is exactly divisible by 5. Step 3: After substituting the remainder in the expression, check it again by the divisor.

Correct answer: B. 6

Explanation: Step 1: In (2a+b), the remainder part is (2\times4+7=15). Step 2: (15=9+6), so the remainder is 6. Step 3: In such questions, handle the remainder of each term separately.

Correct answer: A. 0

Explanation: Step 1: The original remainder is 29. Step 2: 61 leaves remainder 1 when divided by 30, so total remainder (29+1=30), which becomes 0. Step 3: It is useful to reduce a large added number to a smaller remainder first.

Correct answer: B. 1

Explanation: Step 1: Replace (n) by its remainder 2. Step 2: The remainder of (n^2+n+1) comes from (2^2+2+1=7), and 7 leaves remainder 1 when divided by 3. Step 3: In polynomial-like expressions, substituting the remainder makes the solution simple.

Correct answer: A. 13

Explanation: Step 1: The remainder of (a) is 17. Step 2: For (5a), compute (5\times17=85), and (85=24\times3+13). Step 3: After multiplication, the remainder must be reduced below the divisor.

Correct answer: C. 41

Explanation: Step 1: The remainder must satisfy (0\le r<41). Step 2: 41 is equal to the divisor, so it cannot be a remainder. Step 3: In definition-based questions, (r<b) is the most important rule.

Correct answer: A. 0

Explanation: Step 1: The expression (n^2+2n+1) is ((n+1)^2). Step 2: Since (n) has remainder 7, (n+1) has remainder 0, so its square is divisible by 8. Step 3: Recognizing the structure of the expression reduces calculation.

Correct answer: A. 0

Explanation: Step 1: Add the remainders: (13+16+7=36). Step 2: 36 is exactly divisible by 18, so the remainder is 0. Step 3: In multi-term questions, add only the remainders and then reduce at the end.