![]() ![]() Prove that the final number must be less than 2000. Eventually one number is left on the board. Students take turns to replace two numbers a, b from the current list by their sum divided by 2. Problem 108 Suppose real numbers a, b, c, d satisfy a b 0, prove thatĢ = 2 a b a + b ⩽ a b ⩽ a + b 2 ⩽ a 2 + b 2 2 ( HM ⩽ GM ⩽ AM ⩽ QM )Īre written on the board. ![]() (c) Mark on the coordinate plane all points ( x, y) satisfying the inequality (b) Mark on the coordinate plane all points ( x, y) satisfying the inequality (a) Mark on the coordinate plane all points ( x, y) satisfying the inequality Problem 106 Find numbers a and b with the property that the set of solutions of the inequality Problem 105 Mark on the coordinate line all those points x for which (b) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false: (a) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false: Problem 103 Mark on the coordinate line all those points x for which 7.1: Calculator Shortcut for Modular Arithmetic is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. The symbol means congruent to and means that 17 and 2 are equivalent, after you consider the modulus 5. You can type numbers or numerical expressions on the input boxes at the left. X > 1, x > 2, x > 3, x > 4, x > 5, x > 6, x > 7. Sometimes, instead of seeing 17 mod 5 2, you’ll see 17 2 (mod 5). This Web application can solve equations of the form ax² + bx + c 0 (mod n) where the integer unknown x is in the range 0 x < n.In particular, it can find modular square roots by setting a -1, b 0, c number whose root we want to find and n modulus. Problem 102 Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false: What fraction of the interval [0,1) have you marked? (d) Mark on the interval [0,1) all those points x which have a digit “1” in at least one position of their base 3 expansion. (c) Mark on the interval [0,1) all those points x which have a digit “1” in at least one position of their base 2 expansion. (b) Mark on the interval [0,1) all those points x which have the digit “1” in at least one decimal place. Note: “ denotes the interval including both endpoints and (0,1) denotes the interval excluding both endpoints. (a) Mark on the coordinate line all those points x in the interval [0,1) which have the digit “1” immediately after the decimal point in their decimal expansion. \)Ĥ.2.1 Geometrical interpretation of modulus, of inequalities, and of modulus inequalities
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