Question

"Numarul natural "a" are n cifre , iar" a la a 5-a" are m cifre. Stind ca m + n=2019 ,care este valoarea lu "m-n="?"
Va rog, urgent mai am 5 minute!!!

Answer 1

Răspuns:

$\color{cc0000} \boxed{\huge \bf m - n = 1345}$

Explicație pas cu pas:

Salutare!

$\bf {10}^{n - 1} \leqslant a < {10}^{n} \implies {10}^{2n - 5} \leqslant {a}^{5} < {10}^{5n}$

$\bf Dar \: {10}^{m- 1} \leqslant {a}^{5} < {10}^{m} \: \: avem :$

$\bf {10}^{m - 1} \leqslant {a}^{5} < {10}^{n} \implies m - 1 < 5n$

$\bf {10}^{5n - 5} \leqslant {a}^{5} < {10}^{m} \implies 5n - 5< 5m$

$\bf Din \: cele \: de \: mai \: sus \: vom \: avea:5n - 5 < m < 5n + 1 \implies$

$\bf 5n - 5 < 2019 - n < 5n + 1 \implies2018 < 6n < 2024 \: \bigg|:6$

$\bf \implies \boxed{ \large \bf n = 337} \: \: si \: \: \boxed{ \large \bf m = 1682}$

$\color{cc0000} \boxed{ \large \bf m - n = 1682 - 337 = \boxed{\large \bf 1345}}$

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