Let x be the least number which when divided by 32, 48, 90 and 64, the…

When a positive integer x, divided by several different divisors, leaves remainders such that (divisor − remainder) is the SAME constant k for every one of those divisors, x is not built from any single remainder. Then x + k must be a positive common multiple of all the divisors. Therefore, the least such positive x equals the LCM of the divisors minus k, because subtracting k from the least positive common multiple reproduces the required remainder under every divisor.

1. Find the difference between each divisor and its respective remainder: 32 − 20 = 12, 48 − 36 = 12, 90 − 78 = 12, and 64 − 52 = 12. The difference is the same constant, k = 12.

2. Compute the LCM of the divisors 32, 48, 90, and 64 by prime factorization: 32 = 25, 48 = 24 × 3, 90 = 2 × 32 × 5, 64 = 26. Taking the highest power of each prime gives LCM = 26 × 32 × 5 = 2880.

3. Apply the rule from the concept above: the least value of x is LCM − k = 2880 − 12 = 2868.

4. Sum the digits of x: 2 + 8 + 6 + 8 = 24.

Cross-check: dividing 2868 by 32 gives quotient 89 remainder 20; by 48 gives quotient 59 remainder 36; by 90 gives quotient 31 remainder 78; and by 64 gives quotient 44 remainder 52 — all four match the stated remainders, confirming x = 2868 is both valid and minimal.

So the sum of the digits of x is 24.