To find the greatest number that will divide 49, 97, and 189 leaving the same remainder in each case, we need to employ the concept of differences and use the Greatest Common Divisor (GCD) to solve the problem.

1. Let's denote the greatest number by \(x\) and the common remainder by \(r\). This means that:
   • \(49 = k_1 \cdot x + r\)
   • \(97 = k_2 \cdot x + r\)
   • \(189 = k_3 \cdot x + r\)
2. Subtract the first equation from the others to eliminate \(r\):
   • \((97 - 49) = (k_2 - k_1) \cdot x\)
   • \((189 - 97) = (k_3 - k_2) \cdot x\)
   • \((189 - 49) = (k_3 - k_1) \cdot x\)
3. Calculate the differences:
   • \(97 - 49 = 48\)
   • \(189 - 97 = 92\)
   • \(189 - 49 = 140\)
4. The greatest number \(x\) should divide all these differences. Thus, we need to find the GCD of 48, 92, and 140.
5. Compute the GCD:
   • \(\text{GCD(48, 92, 140)}\)
   • First find GCD(48, 92):
     • Prime factorize 48: \(48 = 2^4 \times 3^1\)
     • Prime factorize 92: \(92 = 2^2 \times 23^1\)
     • \(\text{GCD(48, 92) = 2^2 = 4}\)
   • Now find GCD(4, 140):
     • Prime factorize 140: \(140 = 2^2 \times 5^1 \times 7^1\)
     • \(\text{GCD(4, 140) = 4}\)
6. Hence, the GCD of 48, 92, and 140 is \(4\).

Therefore, the greatest number that divides 49, 97, and 189 leaving the same remainder is 4.