A systematic study of the factorization of long-range azimuthal two-particle correlations into a product
of single-particle anisotropies is presented as a function of
p
T
and
η
of both particles and as a function of the
particle multiplicity in PbPb and
p
Pb collisions. The data were taken with the CMS detector for PbPb collisions at
√
s
NN
=
2
.
76 TeV and
p
Pb collisions at
√
s
NN
=
5
.
02 TeV, covering a very wide range of multiplicity. Factorization
isobservedtobebrokenasafunctionofbothparticle
p
T
and
η
. When measured with particles of different
p
T
,
the magnitude of the factorization breakdown for the second Fourier harmonic reaches 20% for very central PbPb
collisions but decreases rapidly as the multiplicity decreases. The data are consistent with viscous hydrodynamic
predictions, which suggest that the effect of factorization breaking is mainly sensitive to the initial-state conditions
rather than to the transport properties (e.g., shear viscosity) of the medium. The factorization breakdown is also
computed with particles of different
η
. The effect is found to be weakest for mid-central PbPb events but becomes
larger for more central or peripheral PbPb collisions, and also for very-high-multiplicity
p
Pb collisions. The
η
-dependent factorization data provide new insights to the longitudinal evolution of the medium formed in heavy
ion collisions.