Using geometric series, one can show that for any expression x^n + y^n where n is an odd number, x+y is a factor. Using this rule, we can easily find factors for this sum.
First, break down the exponent using its factors:
2^105 + 3^105 =
(((2^3)^5)^7) + (((3^3)^5)^7)
If we simplify:
((8^5)^7) + ((27^5)^7)
8^35 + 27^35
According to the rule, 8+27 = 35 is a factor of this sum. Since 35 is divisible by 7, the sum is also divisible by 7.
Now reorder the exponents:
(((2^5)^3)^7) + (((3^5)^3)^7)
32^21 + 243^21
32+243 = 275, which is divisible by both 11 and 25. Therefore, the sum is divisible by 11 and 25.
Comments
Using geometric series, one can show that for any expression x^n + y^n where n is an odd number, x+y is a factor. Using this rule, we can easily find factors for this sum.
First, break down the exponent using its factors:
2^105 + 3^105 =
(((2^3)^5)^7) + (((3^3)^5)^7)
If we simplify:
((8^5)^7) + ((27^5)^7)
8^35 + 27^35
According to the rule, 8+27 = 35 is a factor of this sum. Since 35 is divisible by 7, the sum is also divisible by 7.
Now reorder the exponents:
(((2^5)^3)^7) + (((3^5)^3)^7)
32^21 + 243^21
32+243 = 275, which is divisible by both 11 and 25. Therefore, the sum is divisible by 11 and 25.