1) It is proved that if a divides B × C, and a and B are coprime, then a divides C (ABC is an integer). If the theorem is wrong, give an example and modify it, and prove the modified theorem 2) It is proved that if a and B are positive integers, if a > b, then the square of a > the square of B, and vice versa

1) It is proved that if a divides B × C, and a and B are coprime, then a divides C (ABC is an integer). If the theorem is wrong, give an example and modify it, and prove the modified theorem 2) It is proved that if a and B are positive integers, if a > b, then the square of a > the square of B, and vice versa

prove:
(1) Let BC = Ka, K ∈ Z
Let the prime factor of B be decomposed into
b=p1^x1 × p2^x2 × …… × pr^xr
1, 2 , R are subscripts, ^ X1 represents the power of x1, and P1, P2 are prime numbers, the same below)
a=q1^y1 × q2^y2 × …… × qs^ys
Because (a, b) = 1, so
{p1,p2,p3,…… ,pr}∩{q1,q2,q3,…… , QS} = empty set
From the decomposition of qualitative factors, we can only know that since the qualitative factors of a are not in B, they must all be in C, so there is
A | C, end of syndrome
(2) ∵ a, B are all positive numbers and a > B
∴ |a| > |b|
||a ×|a | > |a |×|b | > |b |×|b |
|A | & sup2; > | B | & sup2;, the end of the certificate

If m, N, N + 1 (m, n are positive integers) can form Pythagorean number, find the relationship between M and n

A:
① If n + 1 is the hypotenuse, then:
M & # 178; + n & # 178; = (n + 1) &# 178;, which is reduced to M = √ (2n + 1)
② If M is the hypotenuse, then:
N & # 178; + (n + 1) &# 178; = M & # 178;, which is reduced to M = √ (2n & # 178; + 2n + 1)

A three digit m and a four digit n are represented by the algebraic expression of M and N, and the seven digit m on the left side of N?
What are the seven digits of m to the right of N?

M is placed on the left, and there are four more numbers in the back, which is equivalent to 10000 times larger
So this seven digit number is:
10000m+n
Similarly, if n is placed around, it is equivalent to 1000 times larger
1000n+m