Euclid’s Division Lemma

Step 1: The remainder of (N) is 39. Step 2: 102 leaves remainder 8 when divided by 47. Step 3: The total remainder is (39+8=47), so the final remainder is 0.

Step 1: When the divisor is 69, the remainder can be from 0 to 68. Step 2: 69 is equal to the divisor, so it cannot be a remainder. Step 3: In definition-based questions, check the condition (r<b) first.

Step 1: (412\times23=9476) and (412\times24=9888). Step 2: The nearest lower multiple below 9876 is 9476, so the remainder is (400). Step 3: Since the remainder is less than 412, it is valid.

Step 1: The remainder of (t) is 37. Step 2: The remainder part of (6t+17) is (6\times37+17=239). Step 3: (239=46\times5+9), so the final remainder is 9.

Step 1: For thirteen times the number, the remainder part is (13\times15=195). Step 2: (195=73\times2+49), so the final remainder is 49. Step 3: After multiplication, reduce the result by the divisor to make a valid remainder.

Step 1: On division by 11, possible remainders are from 0 to 10. Step 2: Eleven consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 11.

Step 1: The remainders in the two forms are 5 and 7. Step 2: (5^2=25) and (7^2=49), and both leave remainder 1 when divided by 12. Step 3: In form-based questions, work only with the remainder.

Step 1: (a-140=79q+61-140=79q-79). Step 2: This can be written as (79(q-1)+0). Step 3: The number is exactly divisible by 79, so the remainder is 0.

Step 1: For the square, take (27^2=729). Step 2: (729=34\times21+15), so the remainder is 15. Step 3: The final answer must always be less than 34.

Step 1: The remainder of (x) is 13. Step 2: The remainder of (x^2+x) comes from (13^2+13=182). Step 3: Since 182 is exactly divisible by 14, the remainder is 0.

Step 1: (157\times43=6751) and (157\times44=6908). Step 2: The nearest lower multiple below 6789 is 6751, so the remainder is (6789-6751=38). Step 3: Do not accept a negative remainder or a remainder greater than 157.

Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (112\times27+111=3024+111=3135). Step 3: The remainder 111 is less than divisor 112, so the form is valid.

Step 1: For the difference, the remainder is (10-23=-13). Step 2: Add 41 to make it a valid remainder, giving 28. Step 3: In subtraction, add the divisor when the remainder becomes negative.

Step 1: On division by 12, possible remainders are from 0 to 11. Step 2: Twelve consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 12.

Step 1: The original remainder is 85. Step 2: 259 leaves remainder 1 when divided by 86. Step 3: The total remainder is (85+1=86), so the final remainder is 0.

Step 1: The remainder of (u) is 13. Step 2: The remainder of (u^2-u) comes from (13^2-13=156). Step 3: (156=20\times7+16), so the remainder is 16.

Step 1: (12^2=144), and 144 leaves remainder 11 when divided by 19. Step 2: For (12^4), check (11^2=121). Step 3: (121=19\times6+7), so the remainder is 7.

Step 1: The original remainder is 37. Step 2: 59 leaves remainder 11 when divided by 48. Step 3: (37+11=48), so the new remainder is 0.

Step 1: The remainders of (a) and (b) are 17 and 22. Step 2: The remainder of (ab+a) comes from (17\times22+17=391). Step 3: (391=24\times16+7), so the final remainder is 7.

Step 1: In standard form, the remainder must be from 0 to 899. Step 2: (900\times10=9000), so (9001=9000+1). Step 3: A negative remainder or a remainder greater than 900 does not make the standard Euclidean form.