4

Part of 2006 Czech-Polish-Slovak Match

Problems(1)

There exists a power of 2 with exactly k consecutive zeroes

Source: Czech-Polish-Slovak 2006 Q4

Show that for every integer

k \ge 1

there is a positive integer

n

such that the decimal representation of

2^n

contains a block of exactly

k

2^n = \dots a00 \dots 0b \cdots

k

a, b \ne 0

modular arithmeticnumber theory unsolvednumber theory