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PDF Editor FAQ

What is A in numbers, if A= 5B+7, and B = 4 (20+A)?

A= 5B+7, and B = 4 (20+A)=> [math]A= 5(4)(20+A)+7[/math][math]=>[/math][math]A=400+20A +7[/math]=> 19A=-407 => [math]\frac{-407}{19}[/math]

AB + 2 = A9; AB x 2 = 9A Question: Which figures represent A and B? (AB is a two-digit number)?

[math]10A+B+2=10A+9[/math][math]B=7[/math][math](10A+7)×2=90+A[/math][math]20A+14=90+A[/math][math]19A=90-14[/math][math]A=76/19[/math][math]A=4[/math][math]Then AB(two-digit number) is 47[/math]

Let x and y be positive integers such that 19x^20 = 17y^19. The minimum value of x has a prime factorisation (a^b) (c^d). What is a+b+c+d?

We are to solve:[math]19n^{20}=17m^{19}[/math]with [math]n,m \in \mathbb{N}[/math].Since [math]\text{gcd}(19,17)=1[/math] we need to have:[math]19|m,\,17|n[/math].It is easy to see that we should choose the following templates:[math]n=17^a 19^b,\, m=17^c 19^d[/math]Substitute into the given equation and derive:[math]17^{20a}19^{20b+1}=17^{19c+1}19^{19d}[/math]We therefore need to have:[math]20a=19c+1,\,20b+1=19d[/math]Or:[math]19a+a=19c+1[/math] giving [math]a=1,\,c=1[/math]and[math]19b+b+1=19d[/math] or [math]19(d-b)=b+1[/math].The smallest value for [math]d-b[/math] equals [math]1[/math].This gives [math]b+1=19[/math] or [math]b=18,\,d=19[/math].We thus have:[math]n=17^{1}19^{18}[/math]The requested value is thus:[math]3\cdot18+1=55[/math]I chose different characters. Sorry… Furthermore you need to add an argument that these choices actually minimize [math]17+19+a+b[/math].

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