  It is proved that there exists an absolute constant c > 0 such that for every
natural number k, every non-bipartite 2-connected graph with average degree at
least ck contains k cycles with consecutive odd lengths. This implies the
existence of the absolute constant d > 0 that every non-bipartite 2-connected
graph with minimum degree at least dk contains cycles of all lengths modulo k,
thus providing an answer (in a strong form) to a question of Thomassen. Both
results are sharp up to the constant factors.
