Find the least number when divided by 7 gives the reminder 6, when divided by…

Step-by-Step Solution

This problem is a classic example involving a constant difference between the divisor and the remainder.

1. Analyze the pattern of remainders:

   • When divided by 7, the remainder is 6 (7 - 6 = 1).

   • When divided by 6, the remainder is 5 (6 - 5 = 1).

   • When divided by 5, the remainder is 4 (5 - 4 = 1).

   • This pattern continues down to 1. In every case, the divisor minus the remainder equals 1.

2. Formulate the logic:

   • Let the number be m.

   • Based on the pattern, m + 1 must be perfectly divisible by all the divisors involved (7, 6, 5, 4, 3, 2, 1).

   • Therefore, m + 1 must be a multiple of the Least Common Multiple (LCM) of these numbers.

3. Calculate the LCM:

   • LCM of (7, 6, 5, 4, 3, 2, 1)

   • The prime factors are: 7, 6 (23), 5, 4 (22), 3, 2, 1.

   • Taking the highest powers of each prime: 7 * 5 * 4 * 3 = 420.

   • So, LCM = 420.

4. Find m:

   • m + 1 = 420

   • m = 420 - 1

   • m = 419.