- #1
bobsmiters
- 12
- 0
If n has k digits in its binary numeral, show that there are at most 2^k/2 numbers n. Can there be exactly 2^k/2?
I tried to understand this question with an example so I took n=36 which has the binary number 100100; k=6 but 2^k/2n gives 2^3 36 but 8 is not less than or equal to 6? Any help is appreciated for either question.
Also does anyone know how to prove this:
Suppose that p_1, p_2, ..., p_k are all the primes that divide a or b, and that a=p_1^m_1 X p_2^m_2 X...X p_k^m_k, b=p_1^n_1 X p_2^n_2 X...Xp_k^n_k
Deduce that: gcd(a,b) = p_1^min(m_1, n_1)Xp_2^min(m_2,n_2)...Xp_k^min(m_k, n_k),
lcm(a,b) = p_1^max(m_1, n_1)Xp_2^max(m_2,n_2)...Xp_k^max(m_k, n_k)
I tried to understand this question with an example so I took n=36 which has the binary number 100100; k=6 but 2^k/2n gives 2^3 36 but 8 is not less than or equal to 6? Any help is appreciated for either question.
Also does anyone know how to prove this:
Suppose that p_1, p_2, ..., p_k are all the primes that divide a or b, and that a=p_1^m_1 X p_2^m_2 X...X p_k^m_k, b=p_1^n_1 X p_2^n_2 X...Xp_k^n_k
Deduce that: gcd(a,b) = p_1^min(m_1, n_1)Xp_2^min(m_2,n_2)...Xp_k^min(m_k, n_k),
lcm(a,b) = p_1^max(m_1, n_1)Xp_2^max(m_2,n_2)...Xp_k^max(m_k, n_k)