Hence or otherwise show that every positive rational number can be exp

1.	Hence or otherwise show that every positive rational number can be expressed in the form a /10b (10c −1) for some natural numbers a, b, c.
Answer» Sol. If p/q is any rational number (p > 0, q > 0), then we may write q = 2^r 5^s t, where t is coprime to 10. Choose a number m having only 1's as its digits and is divisible by t. Consider 9m, Which has only 9 as its digits and is still divisible by t. Let k = 9 m/t. We see that; qk = 9m 2^r 5^s = (10º –1) 2^r 5^s , where c is the number of digits in m. Hence we can find d such that qd = 10^b (10^c – 1) (multiply by a suitable power of 2 if s > r and by a suitable power of 5 if r > s). Then p/q =pd/qd = a/10^b(10^c - 1) where a = pd.

Hence or otherwise show that every positive rational number can be expressed in the form a /10b (10c −1) for some natural numbers a, b, c.

Answer»

Sol.

If p/q is any rational number (p > 0, q > 0), then we may write q = 2^r 5^s t, where t is coprime to 10. Choose a number m having only 1's as its digits and is divisible by t. Consider 9m, Which has only 9 as its digits and is still divisible by t. Let k = 9 m/t. We see that;

qk = 9m 2^r 5^s = (10º –1) 2^r 5^s

, where c is the number of digits in m. Hence we can find d such that qd = 10^b (10^c – 1) (multiply by a suitable power of 2 if s > r and by a suitable power of 5 if r > s). Then

p/q =pd/qd = a/10^b(10^c - 1)

where a = pd.

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