Let a,b,c be positive integers such that a divides b^2 , b divides c^2 and c divides a^2 . Prove that abc divides (a + b + c)^7 .
To prove that abc divides (a + b + c)^7, we can use the concept of prime factorization and the properties of divisibility.
We are given that a divides b^2, b divides c^2, and c divides a^2. This means that the prime factors of a are a subset of the prime factors of b^2, the prime factors of b are a subset of the prime factors of c^2, and the prime factors of c are a subset of the prime factors of a^2.
Let's consider the prime factorization of a, b, and c:
a = p1^a1 * p2^a2 * p3^a3 * ... * pn^an,
b = p1^b1 * p2^b2 * p3^b3 * ... * pn^bn,
c = p1^c1 * p2^c2 * p3^c3 * ... * pn^cn,
where pi are distinct prime factors, ai, bi, and ci are positive integers representing the exponents of the respective prime factors.
Now, let's calculate (a + b + c)^7:
(a + b + c)^7 = [(p1^a1 * p2^a2 * p3^a3 * ... * pn^an) + (p1^b1 * p2^b2 * p3^b3 * ... * pn^bn) + (p1^c1 * p2^c2 * p3^c3 * ... * pn^cn)]^7.
Expanding this expression using the binomial theorem, we get:
(a + b + c)^7 = Sum of [(p1^x1 * p2^x2 * p3^x3 * ... * pn^xn) * (p1^y1 * p2^y2 * p3^y3 * ... * pn^yn) * (p1^z1 * p2^z2 * p3^z3 * ... * pn^zn)],
where xi, yi, and zi are non-negative integers satisfying xi + yi + zi = 7.
Since a divides b^2, b divides c^2, and c divides a^2, we know that the exponents ai, bi, and ci satisfy:
2bi <= ai, 2ci <= bi, 2ai <= ci.
Let's consider one term in the expansion of (a + b + c)^7:
(p1^x1 * p2^x2 * p3^x3 * ... * pn^xn) * (p1^y1 * p2^y2 * p3^y3 * ... * pn^yn) * (p1^z1 * p2^z2 * p3^z3 * ... * pn^zn).
To prove that abc divides (a + b + c)^7, we need to show that the exponents yi, zi, and xi are non-negative integers satisfying yi + zi + xi ≤ bi + ci + ai.
Since 2bi ≤ ai, we have bi ≤ ai/2.
Similarly, bi ≤ ai/2 ≤ ci/4, and ci ≤ ai/4.
Adding these inequalities, we get: bi + ci ≤ ai/2 + ai/4 = 3ai/4.
Therefore, yi + zi + xi ≤ bi + ci + ai ≤ 3ai/4 + ai = 7ai/4.
Since ai is a positive integer (as a is a positive integer), we have 7ai/4 ≤ 7ai.
Hence, yi + zi + xi ≤ bi + ci + ai ≤ 7ai.
This means that the exponents yi, zi, and xi are non-negative integers satisfying yi + zi + xi ≤ bi + ci + ai, which implies that each term in the expansion (a + b + c)^7 is divisible by abc.
Therefore, abc divides (a + b + c)^7.
QED