A positive integer m is created so that its digit sum is 900 and when m...

A positive integer m is created so that its digit sum is 900 and when m is doubled, its digit sum does not change. The digits of m include at least one of each of 3, 4, 5, and 6, and no other digits. Let m+ be the maximum such value of m, let m− be the minimum such value of m, and let P be the result of m+ × m−. Determine the number of digits of P .

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Manojponduru answered 1 year ago

How many eight-digit positive integers have the sum of digits being even?

Use strong induction to show that every positive integer, n, can be written as a sum...

Use strong induction to show that every positive integer, n, can be written as a sum of powers of two: 20, 21, 22, 23, .....

Use strong induction to show that every positive integer n can be written as a sum...

Use strong induction to show that every positive integer n can be written as a sum of distinct powers of two, that is, as a sum of a subset of the integers 2^0 =1, 2^1 = 2, 2^2 = 4, and so on. [Hint: For the inductive step, separately consider the case where k + 1 is even and where it is odd. When it is even, note that (k + 1)/2 is an integer.]

Find a three digit integer in base five that has the order of its digits reversed...

Find a three digit integer in base five that has the order of its digits reversed when multiplied by 2.

Suppose that m is a fixed positive integer. Show that the initial value problem y' =...

Suppose that m is a fixed positive integer. Show that the initial value problem y' = y2m/(2m+1) , y(0) = 0 has infinitely many continuously differentiable solutions. Why does this not contradict Picard’s Theorem?

The least common multiple of nonzero integers a and b is the smallest positive integer m...

The least common multiple of nonzero integers a and b is the smallest positive integer m such that a|m and b|m. It is denoted [a, b], or sometimes [a, b] for short. Prove the following: 1) If a|k and b|k, then [a, b]|k. 2) If gcd(a, b) = 1, then [a, b] =ab 3) If c >0,then [ca, cb] =c·[a, b]. 4) If a >0 and b >0,then [a, b] =ab / gcd(a, b).

7. Let m be a fixed positive integer. (a) Prove that no two among the integers...

7. Let m be a fixed positive integer. (a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m. (b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.

Matlab program Create a function, when given a two-digit integer as input, prints the equivalent English...

Matlab program Create a function, when given a two-digit integer as input, prints the equivalent English phrase. For example, given 39, output thirty-nine. Limit the input numbers to be in the range 21 to 39.

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field. (4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}

Determine, for a given graph G =V,E and a positive integer m ≤ |V |, whether...

Determine, for a given graph G =V,E and a positive integer m ≤ |V |, whether G contains a clique of size m or more. (A clique of size k in a graph is its complete subgraph of k vertices.) Determine, for a given graph G = V,E and a positive integer m ≤ |V |, whether there is a vertex cover of size m or less for G. (A vertex cover of size k for a graph G =...

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