Comments Off on How many 4-digit numbers are divisible by 3 and have 23 as their last two digits??
Answer Let X be your 4 digit number. X = ‘ab23’ = 1000a + 100b + 2*10 + 3 = ( a + b + 2 ) + 999a + 99b + 21 = ( a + b + 2 ) + 3 x ( 333a + 33b + 7 ) Thus, if 3...
Question The polynomial a8x8 +a7x7 + … + a0 has a8 = 1, a7 = -4, a6 = 7 and all its roots positive and real. Find the possible values for a0. Answer Let the roots be xi. We have Sum xi2 = 42 – 2·7 = 2. By Cauchy-Schwarz inequality, we have (x1·1...
Question Prove that if gcd(a_i, m) = 1 for 1 ≤ i ≤ k, then gcd(a_1*a_2*….*a_k, m) = 1. Answer Prove that if gcd(a_i, m) = 1 for 1 ≤ i ≤ k, then gcd(a_1*a_2*….*a_k, m) = 1. We use “proof by contradiction” method. Let d = gcd (a_1*a_2*….*a_k, m). Let p be ANY...
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