Final answer:
To find the least number satisfying the conditions, we calculate the LCM of the divisors (33, 44, 55) and subtract 2. The LCM is 660, so the least number is 658.
Step-by-step explanation:
To find the least number that when divided by 33, 44, and 55 leaves remainders 31, 42, and 53 respectively, we need to recognize that the remainders are 2 less than the divisors. Thus, we can calculate the number by subtracting 2 from the Least Common Multiple (LCM) of the divisors. To find the LCM of 33, 44, and 55, we need to find the prime factorization of each number and then choose the highest powers of all prime factors present.
- Prime factorization of 33 is 3 x 11.
- Prime factorization of 44 is 22 x 11.
- Prime factorization of 55 is 5 x 11.
Taking the highest powers of all prime factors, we get LCM(33, 44, 55) = 22 x 3 x 5 x 11 = 4 x 3 x 5 x 11 = 660. Now, we subtract 2 to get the desired number: 660 - 2 = 658.
Therefore, the least number that satisfies the given conditions is 658.