Which of the following have a non terminating decimal expansion?
77/210
23/8
If pq is a rational number with terminating decimal expansion where p and q are coprimes, then the prime factorisation of q will be in the form of 2n5m, where n and m are non - negative integers.
Consider the fraction 17210.
Here 210 = 2 × 3 × 5 ×7
Hence 210 is not in the form of 2n5m.
Consider the fraction 239.
Here 9 = 3 × 3 = 32
Hence 9 is not in the form of 2n5m.
Consider the fraction 1780.
Here 80 = 2 × 2 × 2 × 2 × 5 = 24×5
Hence 80 is in the form of 2n5m with n = 4 and m = 1.
Consider the fraction 3550=710.
Here 10 = 2 × 5 = 21×51
Hence 10 is in the form of 2n5m with n = 1 and m = 1.
Therefore, 17210 and 239 will have non - terminating decimal expansion.