The greatest number, which divides 2000 and 2200 to leave 22 and 38 respectively as remainders, is: 

Correct option is D

Given:

When 2000 is divided by the greatest number x, the remainder is 22.

When 2200 is divided by the greatest number x, the remainder is 38.

Solution:

Let the greatest number be denoted by x.

2000 ≡ 22(mod x), which means 2000 −22 is divisible by x. So,

2000 − 22 = 1978

Hence, x divides 1978.

2200 ≡ 38(mod x), which means 2200 − 38 is divisible by x. So,

2200 − 38 = 2162

Hence, divides 2162.

To find the greatest common divisor (GCD) of 1978 and 2162 because the greatest number that divides both will be the GCD of these two numbers.

2162 − 1978 = 184

Now, to find the GCD of 1978 and 184:

1978 ÷ 184 ≈ 10

1978 − 10 × 184 = 1978 − 1840 = 138

Now, find the GCD of 184 and 138:

184 − 138 = 46

Now, find the GCD of 138 and 46:

138 ÷ 46 = 3(quotient)

138 − 3 × 46 = 138 − 138 = 0

Since the remainder is 0, the GCD is 46.

The greatest number that divides 2000 and 2200 to leave remainders of 22 and 38, respectively, is 46.