Prove that every integer greater than 27 can be written as 5a+8b, where a,b∈Z+

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nitruraviX · Accepted Answer

1.Statement: P(n)For n∈N there exist a,b &amp;gt; 0 such thatn+27 = a * 5 + b * 8We use mathematical induction to show that every integer greater than 27 can be written asa*5+b*8, for a,b&amp;gt;0Base Case: Let n = 1. Note that28 = 4 * 5 + 1 * 8Thus base case is true.Induction Hypothesis: Let P(n) is true for all n≤k. That is there exist a,b &amp;gt; 0 such thatk+27=a*5+b*8We have to find a&#039;b&#039; &amp;gt; 0 such that(k+1)+27=a*5+b*82.Inductive Steps: First note that if n = a* 5 + b*8 for n &amp;gt; 27 then either a≥3orb≥3. Otherwise note that if a &amp;lt; 2 and b &amp;lt; 2 thena*5+b*8 &amp;lt; 2*5+2*8=26, a contradiction.Case 1:First assume b≥3 Now by using induction hypothesis we get(k+1)+27=(k+27)+1=(k+27)+(5*5-3*8)=(a*5+b*8)+(5*5-3*8)=(a+5)*5+(b-3)*8In this case we get a&#039; = a+5&amp;gt;0 and b&#039; = b - 3&amp;gt;0.Case 2: Now assume a≥3. Then we consider following derivation(k+1)+27=(k=27)+1=(k+27)+(-3*5=2*8)=(a*5+b*8)+(-3*5+2*8)=(a-3)*5+(b+2)*8In this case we consider a&#039; = a-3&amp;gt;0 and b&#039; = b + 2&amp;gt;0.Result: For n∈N there exist a,b &amp;gt; 0 such thatn+27 = a*5+b*8

Prove that every integer greater than 27 can be written as 5a+8b, where a,b in ZZ^+

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