Select the option that is related to the third number in the same way as the second number… — State PSC Exam

Select the option that is related to the third number in the same way as the second number is related to the first number.

$13 : 160 :: 17 : ?$

320

The relationship follows the dual-operation formula

(N - 1)^2 + (\text{digit sum of } N)^2

. For

13

(13-1)^2 + (1+3)^2 = 144 + 16 = 160

. Applying this to

17

gives

(17-1)^2 + (1+7)^2 = 16^2 + 8^2 = 256 + 64 = 320

Analyze the relationship between the first pair of numbers (

13

and

160

Identify the multi-step pattern:

(x - 1)^2 + (\text{sum of digits of } x)^2

For

x = 13

, the predecessor is

13 - 1 = 12

, and

12^2 = 144

. The sum of the digits of

13

1 + 3 = 4

, and

4^2 = 16

. Adding these gives

144 + 16 = 160

Apply the identical multi-step rule to the third number (

x = 17

Find the squared predecessor and the squared digit sum for

17

The predecessor of

17

17 - 1 = 16

, and

16^2 = 256

. The sum of the digits of

17

1 + 7 = 8

, and

8^2 = 64

Sum the resulting values to determine the missing term.

256 + 64 = 320

Combining the squared predecessor and squared digit sum yields

320

Multi-Step Numerical Analogy based on Predecessor Squares and Digit-Sum Operations

Question