Final answer:
The smallest number that fits the conditions is 1394, which is derived from the LCM of the divisors (20, 25, 35, 40) minus 6.
Step-by-step explanation:
The smallest number which when divided by 20, 25, 35, and 40 leaves a remainder of 14, 19, 29, and 34 respectively is found by calculating the least common multiple (LCM) of the divisors and adjusting for the remainders.
Since the remainders are less by 6, 6, 6, and 6 when compared to the divisors, it means that the number sought, when divided by 20, 25, 35, or 40, is lacking 6 to get a complete division with no remainder.
Therefore, we first find the LCM of the divisors and then subtract 6 to get the desired number.
The LCM of 20, 25, 35, and 40 is 1400. Subtracting 6 gives us 1394.
This means 1394 divided by 20 leaves a remainder of 14, by 25 leaves 19, by 35 leaves 29, and by 40 leaves 34.