Find a common factor of (527 to the power 379 + 473 to the power 379) and (527 to the power 166 - 473 to the power 166)
To find the common factor of (527^379 + 473^379) and (527^166 - 473^166), we can use the method of difference of squares.
Step 1: Let's rewrite the first expression as a difference of squares:
(527^379 + 473^379) = [(527^189)^2 - (473^189)^2]
Step 2: Now, let's rewrite the second expression as a difference of squares:
(527^166 - 473^166) = [(527^83)^2 - (473^83)^2]
Step 3: We can see that both expressions can be factorized as a difference of squares.
(527^379 + 473^379) = [(527^189 + 473^189)(527^189 - 473^189)]
(527^166 - 473^166) = [(527^83 + 473^83)(527^83 - 473^83)]
Step 4: Now, we can see that both expressions have the factor (527^83 - 473^83) in common.
Therefore, the common factor of (527^379 + 473^379) and (527^166 - 473^166) is (527^83 - 473^83).
To find the common factors of two expressions, we need to factorize both expressions first. Let's start with the first expression, (527^379 + 473^379).
To factorize this expression, we notice that both terms have a similar structure (with different bases), and we can use the formula for the sum of two cubes:
a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Applying this formula, we have:
527^379 + 473^379 = (527 + 473)(527^2 - 527 * 473 + 473^2).
Simplifying this expression, we get:
527^379 + 473^379 = 1000(527^2 - 527 * 473 + 473^2).
Now, let's factorize the second expression, (527^166 - 473^166).
Similarly, we can use the formula for the difference of two cubes:
a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Applying this formula, we have:
527^166 - 473^166 = (527 - 473)(527^2 + 527 * 473 + 473^2).
Simplifying this expression, we get:
527^166 - 473^166 = 54(527^2 + 527 * 473 + 473^2).
Now that both expressions are factorized, we can clearly see that the common factor is 54.
Therefore, the common factor of (527^379 + 473^379) and (527^166 - 473^166) is 54.
(X^n - Y^n) is divisible by both ( x+a)& ( x - a ) if n is even,
Please be clear.
did you know that for
x^n + y^n, x + y is a factor if n is odd and for
x^n - y^n, x-y is a factor in n is even
e.g. try some small numbers that fit into your calculator for testing
5^7 + 11^7 = 19565296 and
19565296 divides by 16 evenly 1222831 times
13^6 - 5^6 = 4811184 and
4811184 divides by 8 evenly 601398 times
I will use this property
527379 + 473379
= (527+473)(some big number)
= 1000(some big number)
527166 - 473166
= (527-473)(some other big number)
= 54(some other big number)
so the first one divides by 1000, the other by 54,
so a common factor is 2
note it said to find "a common" factor, not the highest common factor.
There may be others