To determine the sum of terms shared by the sequences \( a_n = 46 + 8n \) and \( b_n = 98 + 4n \) for \( n \leq 100 \), we equate the general terms: \( 46 + 8n = 98 + 4m \).
This simplifies to \( 8n - 4m = 52 \), which further reduces to \( 2n - m = 13 \), or \( m = 2n - 13 \).
Given that \( n \) is a natural number such that \( 1 \leq n \leq 100 \), we find the values of \( n \) for which \( m \) is also a natural number. The condition for \( m \) is \( 1 \leq m \leq 100 \). Substituting \( m = 2n - 13 \), we get \( 1 \leq 2n - 13 \leq 100 \). Solving this inequality for \( n \), we find \( 14 \leq 2n \leq 113 \), which means \( 7 \leq n \leq 56 \).
The common terms are generated by \( a_n = 46 + 8n \) for \( n \) ranging from 7 to 56.
The first common term is \( a_7 = 46 + 8 \cdot 7 = 102 \).
The last common term is \( a_{56} = 46 + 8 \cdot 56 = 494 \).
The total number of common terms is \( 56 - 7 + 1 = 50 \).
The sum of these common terms, forming an arithmetic sequence, is calculated as \( S = \frac{\text{number of terms}}{2}(\text{first term} + \text{last term}) = \frac{50}{2}(102 + 494) = 25 \cdot 596 = 14900 \).
The sum of all terms common to both sequences is \( \boxed{14900} \).