The largest number?

Let: x be the divisor; y be the quotient when 398 is divided by x; z be the quotient when 606 is divided by x; b be the quotient when 474 is divided by x.

$398 = xy + 7$

$606 = xz + 11$

$474 = xb + 15$

Adding these 3 equations,

$1478 = x(y+z+b) +33$

$x(y+z+b) = 1455$

$1455=5\times 289 = 5\times 17\times 17= 17\times 35$

Clearly, 35 cannot be the answer. Therefore, $\boxed{17}$ is the answer.