Several formula theorems and proofs
1. If \ (a \ mid b \) and \ (b \ mid c \) , then \ (a \ mid c \)
Second, if \ (a \ mid b \) and \ (a \ mid c \) , then for any integer \ (x \) , \ (y \) , there is \ (a \ mid (bx + cy) \ ) .
3. Let \ (m \ not = 0 \) , then \ (a \ mid b \) is equivalent to \ (ma \ mid mb \) .
4. Let \ (x \) and \ (y \) satisfy: \ (ax + by = 1 \) , and \ (a \ mid n \) and \ (b \ mid n \) , then \ (ab \ mid n \) .
By \ (ax + by = 1 \) there must be gcd (a, b) = 1, divided into two cases.
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1 \ (a \) \ (b \) are prime numbers, we can directly derive \ (ab \ mid n \) .
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2 One of \ (a \) \ (b \) is 1, we can also directly derive \ (ab \ mid n \) .
Fifth, if \ (QD = B + C \) , then the \ (d \ mid b \) necessary and sufficient condition that \ (D \ C MID \) .
Sufficient condition \ (d \ mid b \ rightarrow d \ mid c \)
Divide both sides of the equation by d at the same time to get \ (z = z + c \ div d \) , if both sides of the equation are integers then \ (c \ div d \) \ (\ epsilon \) \ (z \) , There is \ (d \ mid c \) , where \ (z \) is an integer.
Necessary conditions \ (d \ mid b \ leftarrow d \ mid c \)
Divide by d above to get \ (b \ div d = z + z \) , if both sides of the equation are integers then \ (b \ div d \) \ (\ epsilon \) \ (z \) , there is \ (d \ mid b \) .