If $$\mathrm{(L)_M}$$​ represents a number L in the base-M number system, then which of the following numbers are equivalent to $$(19)_{16}$$​?
A. $$(11001)_{2}$$​​

B. $$(201)_3$$​​

C. $$(121)_4$$​​

D. $$(101)_5$$​​

E. $$(25)_{10}$$​​

Choose the correct answer from the options given below: ​​

To solve the problem, we need to convert $$\bf{(19)_{16}}$$​ (a number in hexadecimal or base-16) into different bases and match it with the options given.

Step 1: Convert $$\bf{(19)_{16}}$$​ to decimal (base 10)
The number 19 in hexadecimal (base 16) can be converted to decimal by expanding the number using powers of 16:
​$$(19)_{16} = (1 \times 16^1) + (9 \times 16^0)$$​​
​$$= 16 + 9 = 25$$​​
So, $$\bf{(19)_{16}}$$​ is 25 in decimal.
Step 2: Check which of the options are equivalent to 25 in decimal:
A. $$\bf{(11001)_2}$$​ (Binary)
To check if $$\bf{(11001)_2}$$​ equals 25 in decimal, convert it to decimal:
​$$(11001)_2 = (1 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)$$​​
​$$= 16 + 8 + 0 + 0 + 1 = 25$$​​
So, $$\bf{(11001)_2} = 25$$​ in decimal.
B. $$\bf{(201)_3}$$​ (Ternary)
To check if $$\bf{(201)_3}$$​ equals 25 in decimal, convert it to decimal:
​$$(201)_3 = (2 \times 3^2) + (0 \times 3^1) + (1 \times 3^0)$$​​
​$$= 2 \times 9 + 0 \times 3 + 1 \times 1$$​​
​$$= 18 + 0 + 1$$​​
​$$= 19$$​​
So, $$\bf{(201)_3} ≠ 25$$​ in decimal.
C. $$\bf{(121)_4}$$​ (Quaternary)
To check if $$\bf{(121)_4}$$​ equals 25 in decimal, convert it to decimal:
​$$(121)_4 = (1 \times 4^2) + (2 \times 4^1) + (1 \times 4^0)$$​​
​$$= 1 \times 16 + 2 \times 4 + 1 \times 1$$​​
​$$= 16 + 8 + 1$$​​
​$$= 25$$​​
So, $$\bf{(121)_4} = 25$$​ in decimal.
D. $$\bf{(101)_5}$$​ (Quinary)
To check if $$\bf{(101)_5}$$​ equals 25 in decimal, convert it to decimal:
​$$(101)_5 = (1 \times 5^2) + (0 \times 5^1) + (1 \times 5^0)$$​​
​$$= 1 \times 25 + 0 \times 5 + 1 \times 1$$​​
​$$= 25 + 0 + 1$$​​
​$$= 26$$​​
So, $$\bf{(101)_5} ≠ 25$$​ in decimal.
E. $$\bf{(25)_{10}}$$​ (Decimal)
This is already in decimal form, so $$\bf{(25)_{10}} = 25$$​.
Step 3: Conclusion
The numbers $$\bf{(11001)_2}, \bf{(121)_4}$$​ and $$\bf{(25)_{10}}$$​ are equivalent to $$\bf{(19)_{16}}$$​.
Thus, the correct answer is:
(a) A, C and E only.
Information Booster:
1. Binary (Base-2): Uses digits 0 and 1, with each digit representing increasing powers of 2.
2. Ternary (Base-3): Uses digits 0, 1 and 2, with each digit representing increasing powers of 3.
3. Quaternary (Base-4): Uses digits 0 to 3, with each digit representing increasing powers of 4.
4. Quinary (Base-5): Uses digits 0 to 4, with each digit representing increasing powers of 5.