The greatest number, which divides 1163 and 2214 to leave 15 and 82 respectively as remainders, is:

Correct option is D

Given:
The greatest number divides 1163 and 2214, leaving remainders 15 and 82 respectively.
Formula Used:
Required greatest number = HCF(Number₁ - Remainder₁, Number₂ - Remainder₂  )
Solution:
Number₁ = 1163, Remainder₁ = 15
Number₂ = 2214, Remainder₂ = 82
New values: 1163 − 15 = 1148 and 2214 − 82 = 2132
Now find HCF(1148, 2132) using step-by-step division method:
2132 ÷ 1148 => Quotient = 1, Remainder = 984
1148 ÷ 984 => Quotient = 1, Remainder = 164
984 ÷ 164 => Quotient = 6, Remainder = 0
The greatest number is 164.