Understanding the Number Series Pattern

The question asks us to find the next number in the given series:

168, 131, 71, 223, 211, 51, 91, 62, 105, ?

To solve number series questions, we look for a pattern in the sequence. This pattern can involve addition, subtraction, multiplication, division, squares, cubes, or operations on the digits of the numbers.

Analyzing the Pattern using Sum of Digits

Let's examine the relationship between each number and the sum of its digits. Let $S(N)$ denote the sum of the digits of a number $N$. For example, $S(168) = 1+6+8 = 15$.

Let's calculate the sum of digits for each number in the given series:

• $S(168) = 1 + 6 + 8 = 15$
• $S(131) = 1 + 3 + 1 = 5$
• $S(71) = 7 + 1 = 8$
• $S(223) = 2 + 2 + 3 = 7$
• $S(211) = 2 + 1 + 1 = 4$
• $S(51) = 5 + 1 = 6$
• $S(91) = 9 + 1 = 10$
• $S(62) = 6 + 2 = 8$
• $S(105) = 1 + 0 + 5 = 6$

The sequence of the sum of digits is: 15, 5, 8, 7, 4, 6, 10, 8, 6.

Discovering the Core Pattern: Number minus Sum of Digits

Let's try calculating the difference between each number and the sum of its digits ($N - S(N)$):

• $168 - S(168) = 168 - 15 = 153$
• $131 - S(131) = 131 - 5 = 126$
• $71 - S(71) = 71 - 8 = 63$
• $223 - S(223) = 223 - 7 = 216$
• $211 - S(211) = 211 - 4 = 207$
• $51 - S(51) = 51 - 6 = 45$
• $91 - S(91) = 91 - 10 = 81$
• $62 - S(62) = 62 - 8 = 54$
• $105 - S(105) = 105 - 6 = 99$

This gives us a new sequence:

153, 126, 63, 216, 207, 45, 81, 54, 99

Identifying the Pattern in the Derived Sequence

Let's look for a pattern in the sequence 153, 126, 63, 216, 207, 45, 81, 54, 99. This sequence has 9 terms, corresponding to the 9 numbers in the original series before the question mark.

Upon careful observation, the first term in this sequence is 153.

Let's assume that this sequence derived from the original series repeats, meaning the 10th term in this derived sequence is the same as the 1st term.

The 10th term in the original series is the question mark (?). Let's call this number $X$. The 10th term in the derived sequence would be $X - S(X)$.

According to our assumption, the 10th term in the derived sequence is equal to the 1st term, which is 153.

So, we must have: $X - S(X) = 153$.

Finding the Missing Number from Options

We need to find which of the given options satisfies the equation $X - S(X) = 153$. Let's test each option:

Option 1: 135

• $X = 135$
• $S(135) = 1 + 3 + 5 = 9$
• $X - S(X) = 135 - 9 = 126$
• $126 \ne 153$. So, 135 is not the answer.

Option 2: 187

• $X = 187$
• $S(187) = 1 + 8 + 7 = 16$
• $X - S(X) = 187 - 16 = 171$
• $171 \ne 153$. So, 187 is not the answer.

Option 3: 165

• $X = 165$
• $S(165) = 1 + 6 + 5 = 12$
• $X - S(X) = 165 - 12 = 153$
• $153 = 153$. This matches! So, 165 is the answer.

Option 4: 137

• $X = 137$
• $S(137) = 1 + 3 + 7 = 11$
• $X - S(X) = 137 - 11 = 126$
• $126 \ne 153$. So, 137 is not the answer.

The number that comes in place of the question mark is 165, as it satisfies the pattern $X - S(X) = 153$.

Conclusion

The pattern in the series is based on the value obtained by subtracting the sum of its digits from the number itself. This derived sequence seems to repeat. The missing number is 165.