It is known that the greatest common divisor of a and b is 14, the least common multiple of a and c is 350, and the least common multiple of b and c is also 350. How many groups of positive integers a, b, c satisfy the above conditions?
已知a与b的最大公约数是14，a与c的最小公倍数是350，b与c的最小公倍数也是350．满足上述条件的正整数a、b、c共有多少组？
<gadget id="calculator">5 * 4</gadget>
<output>20</output>

<result>20</result>