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From an article by Chinese mathematician LooKeng-Hua:
"Someone can blurt out that the cube root of 188,132,517 is 573. Of course, 188 determines the first digit 5, and the last digit 7 determines the 3, but the reader might wonder, how is the middle 7 calculated? It can be seen that there are two methods: one from the beginning to the end, and one from the end to the beginning."

I cannot understand what is the method to calculate the middle digit 7 by mental arithmetic.

Bill Dubuque
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nick7226
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2 Answers2

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Cast elevens $\Rightarrow\bmod 11\!:\ \color{#0a0}{188132517\equiv 1}\equiv (\color{#c00}{503}\!+\!10t)^3\equiv (\color{#c00}8\!-\!t)^3\,$ so $\,8\!-\!t\equiv \color{#0a0}1\,$ so $\,t\equiv 7\ \ $

Bill Dubuque
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  • by $,\color{#0a0}{188132517}\equiv 1!-!8!+!8!-!1!+!3!-!2!+!5!-!1!+!7\equiv \color{#0a0}1,,\ \color{#c00}{503}\equiv 5!-!0!+!3\equiv \color{#c00}8\pmod{!11}\ \ $ – Bill Dubuque Oct 05 '24 at 08:20
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    Please strive not to answer PSQs. – Sahaj Oct 05 '24 at 08:38
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    @Sahaj Showing working is not the only way to demonstrate context. The source and the motivation for this question are perfectly clear. – Toby M Oct 05 '24 at 15:14
  • $!!\bmod 11!:\ \color{#c00}{x^3}\equiv\color{#0a0} 1\Rightarrow x\equiv \color{#0a0}1,$ by $,x\equiv{x^{10}/(\color{#c00}{x^3})^3}!\equiv 1/\color{#c00}1^3.,$ Or by $:!\color{#c00}{3}$'rd roots unique, by $,\gcd(\color{#c00}3,\underbrace{\phi(11)}_{\large 10})!=!1.,$ – Bill Dubuque Oct 05 '24 at 21:05
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    @TobyM There is nothing in the question save for a quote, and a plea close to "I have no clue. Help me". zero context. Sahaj was spot on in his reminder to a user who barks the same to countless answerers. – amWhy Oct 20 '24 at 23:11
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<< Too long for comment >>

When we learn (memorize) the multiplication table in school , we are targeting mental arithmetic. That gets extended to this type of advanced calculations , which naturally requires advanced memorization.

When we know what we have is $(500+10X+3)^3$ , we will use the memorized values for :
$503^3=\color{gray}{127263527}$
$(500+10X+3)^3=1000 X^3 + \color{violet}{150900} X^2 + \color{violet}{7590270} X + 127263527$ (just like the multiplication table , the Sum of the two numbers highlighted might be required to be memorized too)

It is then just a matter of guesstimating $[188132517-\color{gray}{127263527}]/[\color{violet}{150900}+\color{violet}{7590270}]=60868990/7741170 \approx 609/77 \approx 7.91$

<< Middle Digit is $7$ >>

ADDENDUM :

There is no "one over-all method" to make such calculations.
Each Case & Each "Expert" will have a Set of techniques. I have high-lighted just one technique here.

Prem
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