Divisibility Properties of the Sequence 10\(^n\) + 1 with Shared Prime Factors Where n ∈ Z\(^+\)
ANGELA MAE E. BALAGAPO
College of Science, University of Eastern Philippines, Philippines.
JESSICA RAY E. DEL CORRO
College of Science, University of Eastern Philippines, Philippines.
SHIELA MARIE P. GALERO
College of Science, University of Eastern Philippines, Philippines.
JHANINE M. MAHILUM *
College of Science, University of Eastern Philippines, Philippines.
DANILO C. BASISTA
College of Science, University of Eastern Philippines, Philippines.
BEA ELAINE F. PEDAMATO
College of Science, University of Eastern Philippines, Philippines.
*Author to whom correspondence should be addressed.
Abstract
Prime factorization has been a central theme in number theory for centuries, providing significant insights into the structure and relationships of integers. This study investigated the divisibility properties of the sequence 10n+ 1, where n ∈ Z+ , with an emphasis on identifying and proving patterns involving recurring prime factors. Using the Calces, a scientific calculator application for computational verification, and applying mathematical induction and modular arithmetic for rigorous proofs, the study establishes four key divisibility properties:
i. 10n+ 1 is divisible by 11 for n = 2m - 1, where m ∈ Z+
ii. 10n+ 1 is divisible by 101 for n = 4m - 1, where m ∈ Z+
iii. 10n+ 1 is divisible by both 73 and 137 for 11 for n = 8m - 4 , where m ∈ Z+
iv. 10n+ 1 is divisible by 7, 11 and 13 for n = 6m - 3 , where m ∈ Z+
These results reveal regular, modular-based prime factorizations of the form 10n+ 1, where n ∈ Z+, uncovering deep patterns and predictable divisibility behaviors within this exponential sequence. The findings contribute to a broader understanding of number-theoretic structures, particularly in relation to periodic prime factors across integer sequences.
Keywords: Divisibility, positive integers, prime factorization, sequences