  Period change studies give a window to probe into the evolution and dynamics
of Cepheids. While evolutionary period changes have been well studied both
observationally and theoretically, non-evolutionary period changes lack a
systematic and quantitative description. The overall objective is to have a
quantitative understanding of the full picture of non-evolutionary period
changes in Cepheids, to develop a formalism to disentangle it from the secular
evolutionary period change. In the first part of the series of works, we aim to
conduct a systematic search for non-evolutionary period changes to search for
Cepheids in likely binary configuration and quantify their incidence rates in
the Magellanic Clouds. We collect more than decade-long time-series photometry
from the publically available survey, Optical Gravitational Lensing Experiment
(OGLE), with more than 7200 Cepheids collectively from the Large Magellanic
Cloud (LMC) and Small Magellanic Cloud (SMC). Our sample contains both
fundamental-mode and first overtone-mode Cepheids. Then we calculate observed
minus calculated ($O-C$) diagrams to reveal the light-travel time effect
(LTTE). In our search, out of an overall sample of more than 7200 Cepheids, we
found 52 candidate Cepheid binary systems in the LMC (30 fundamental and 22
first overtone-mode) and 145 in the SMC (85 fundamental and 60 first
overtone-mode). The majority of the sample is characterized by orbital periods
of 2000-4000\,d and eccentricities of 0.2-0.5. Moreover, we report two
candidates in each galaxy with the Cepheid likely existing with a giant
companion. The incidence rate ratio for SMC to LMC calculated from our sample
is in agreement with binary Cepheid population synthesis predictions.
