Solve the following congruence. Make sure that the number you write is in the range [0,M−1] where M is the modulus of the congruence. If there is more than one solution, write the answer as a list separated by commas. If there is no answer, write N. 180x=276 (mod399)

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Solve the following congruence. Make sure that the number you write is in the range [0,M−1] where M is the modulus of the congruence. If there is more than one solution, write the answer as a list separated by commas. If there is no answer, write N.  180x=276 (mod399)

cheekabooy · Accepted Answer

180x =276(mod399)  It can be seen that ged (180.399) =3 and 3 divides 276, hence  180x =276(mod399) have an integer solution  This can be written as 180x+399y=276  Solve using Euclidian algorithm  399=180×2+39...(i)  180=39×4+24...(ii)  39=24×1+15...(iii)  24=15×1+9...(iv)  15=9×1+6...(v)  9=6×1+3 (Stop here ∵gcd=3)  Now  3=9−6×1  3=9−(15−9×1)×1 (From (v))  3=9×(2)−(15)×1  3=(24−15×1)×(2)−(15)×1 (From (iv))  3=(24)×(2)−(15)×3  3=(24)×(2)−(39−24×1)×3 (From (iii))  3=(24)×(5)−(39)×3  3=(180−39×4)×(5)−(39)×3 (From (ii))  3=(180)×(5)−(39)×23  3=(180)×(5)−(399−180×2)×23 (From (i))  3=(180)×(51)−(399)×23  multiply by 92  276=(180)×(4692)+(399)×(−2116)  Compare it with 180x + 399y = 276  Hence x = 4692  Since 4692∉[0,338]  Hence answer is N

Solve the following congruence. Make sure that the number you write is in the range [0,M−1] where M is the modulus of the congruence. If there is more than one solution, write the answer as a list separated by commas. If there is no answer, write N. 180x=276 (mod399)

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