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Question Number 156775 by mnjuly1970 last updated on 15/Oct/21

$$$$$$fisafunctionthatassigns$$$$toeachnaturalnumber$$$$multiplicationofitsdigits$$$$forexampl:f\left(200\right)=0$$$$f\left(128\right)=1\times 2\times 8=16$$$$findthevalueof:$$$$f\left(100\right)+f\left(101\right)+f\left(102\right)+...f\left(999\right)=?$$$$$$

Commented by Rasheed.Sindhi last updated on 15/Oct/21

$$\underset{a=1,b=0,c=0}{\sum ^{a=9,b=9,c=9}}abc=91125$$

Answered by Rasheed.Sindhi last updated on 15/Oct/21

$$100⩽\overline{abc}⩽999$$$$c=0,1,2,...,9$$$$\overline{ab0},\overline{ab1},\overline{ab2},...\overline{ab9}$$$$f\left(\overline{ab0}\right)+f\left(\overline{ab1}\right)+f\left(\overline{ab2}\right)+,...+f\left(\overline{ab9}\right)$$$$=a.b.0+a.b.1+a.b.2+...+a.b.9$$$$=ab(0+1+2+3+...+9)$$$$=ab\left(\frac{9\times 10}{2}\right)=ab\left(45\right)$$$$b=0,1,2,...,9$$$$a.b\left(45\right):$$$$45a.0+45a.1+45a.2+...+45a.9$$$$=45a(0+1+2+...+9)={45}^{2}a$$$$a=1,2,3,...,9$$$${45}^{2}a:$$$${45}^2.1+{45}^2.2+{45}^2.3+...+{45}^2.9$$$${45}^2(1+2+3+...+9)={45}^3=91125$$$$\mathcal{B}ymyson,\mathrm{Arshad}\mathrm{Soomro}$$$$$$

Commented by mnjuly1970 last updated on 16/Oct/21

$$verynicethanksalot...$$

Commented by qaz last updated on 15/Oct/21

$$\mathrm{S}=\underset{\mathrm{a}=1}{\sum ^9}\underset{\mathrm{b}=1}{\sum ^9}\underset{\mathrm{c}=1}{\sum ^9}\mathrm{abc}=\underset{\mathrm{a}=1}{\sum ^9}\mathrm{a}\underset{\mathrm{b}=1}{\sum ^9}\mathrm{b}\underset{\mathrm{c}=1}{\sum ^9}\mathrm{c}={\left(\frac{9\cdot 10}{2}\right)}^3=91125$$

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