+11 Given that a is a multiple of 456, find the greatest common divisor of 3a^3+a^2+4a+57 and a.
+195 We can start by factoring out an "a" from the polynomial 3a^3 + a^2 + 4a + 57 to get:
3a^3 + a^2 + 4a + 57 = a(3a^2 + a + 4) + 57
Since we know that "a" is a multiple of 456, we can write "a" as "456k" for some integer "k". Substituting this into the above expression, we get:
a(3a^2 + a + 4) + 57 = 456k(3(456k)^2 + 456k + 4) + 57
Simplifying this expression, we get:
a(3a^2 + a + 4) + 57 = 198262272k^3 + 81629312k^2 + 182016k + 57
Now we need to find the greatest common divisor of "a" (which is equal to "456k") and the above expression.
We can use the Euclidean algorithm to find the greatest common divisor.
First, we take the remainder of the expression divided by "a":
198262272k^3 + 81629312k^2 + 182016k + 57 mod 456k
= (198262272k^3 + 81629312k^2 + 182016k) - (435k)(456k) + 57
= 198262272k^3 + 81629312k^2 + 182016k - 197520k + 57
= 198262272k^3 + 81629312k^2 - 146504k + 57
Next, we take the remainder of "a" divided by this expression:
456k mod (198262272k^3 + 81629312k^2 - 146504k + 57)
= (456k) - (198262272k^3 + 81629312k^2 - 146504k + 57)(a')
= 456k - (198262272k^3 + 81629312k^2 - 146504k)(a') - 57a'
= 456k - 198262272k^3a' - 81629312k^2a' + 146504ka' - 57a'
where a' is some integer.
We can repeat the above steps until we get a remainder of 0.
Since "a" is a multiple of 456, the greatest common divisor of "a" and the polynomial 3a^3 + a^2 + 4a + 57 is also 456.
+195 I apologize for the mistake. Let me try to solve the problem again.
Given that "a" is a multiple of 456, we can write "a" as "456k" for some integer "k". Substituting this into the polynomial 3a^3 + a^2 + 4a + 57, we get:
3a^3 + a^2 + 4a + 57 = 3(456k)^3 + (456k)^2 + 4(456k) + 57
Simplifying this expression, we get:
3a^3 + a^2 + 4a + 57 = 198262272k^3 + 81629312k^2 + 182016k + 57
To find the greatest common divisor of this polynomial and "a" (which is equal to "456k"), we can use the Euclidean algorithm.
First, we take the remainder of the polynomial divided by "a":
198262272k^3 + 81629312k^2 + 182016k + 57 mod 456k
= (198262272k^3 + 81629312k^2 + 182016k) - (435k)(456k) + 57
= 198262272k^3 + 81629312k^2 + 182016k - 197520k + 57
= 198262272k^3 + 81629312k^2 - 146504k + 57
Next, we take the remainder of "a" divided by this expression:
456k mod (198262272k^3 + 81629312k^2 - 146504k + 57)
= (456k) - (198262272k^3 + 81629312k^2 - 146504k + 57)(a')
= 456k - 198262272k^3a' - 81629312k^2a' + 146504ka' - 57a'
where a' is some integer.
We can repeat the above steps until we get a remainder of 0.
Since the greatest common divisor of "a" and the polynomial is equal to the remainder when we get a remainder of 0, we can stop here and conclude that the greatest common divisor of 3a^3 + a^2 + 4a + 57 and "a" (which is a multiple of 456) is 456.
I hope this answer is helpful!
