Question

The sum of letters in the word MATHS = 2002
Find the base (less than 10) and the number for each letter. Assuming a set of twenty six consecutive non-negative integers represent letter of alphabet

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{givn}\mathrm{that}, \\ \mathrm{M}+\mathrm{A}+\mathrm{T}+\mathrm{H}+\mathrm{S}=2002 \\ \mathrm{Note}\mathrm{that}, \\ {\left(2002\right)}_3=2\times 3^3+0\times 3^2+0\times 3^1+2\times 3^0=56 \\ {\left(2002\right)}_4=2\times 4^3+0\times 4^2+0\times 4^1+2\times 4^0=130 \\ {\left(2002\right)}_5=2\times 5^3+0\times 5^2+0\times 5^1+2\times 5^0=252 \\ {\left(2002\right)}_6=2\times 6^3+0\times 6^2+0\times 6^1+2\times 6^0=434 \\ {\left(2002\right)}_7=2\times 7^3+0\times 7^2+0\times 7^1+2\times 7^0=688 \\ {\left(2002\right)}_8=2\times 8^3+0\times 8^2+0\times 8^1+2\times 8^0=1026 \\ {\left(2002\right)}_9=2\times 9^3+0\times 9^2+0\times 9^1+2\times 9^0=1460 \\ \mathrm{Since}\mathrm{it}\mathrm{is}\mathrm{assumed}\mathrm{that}\mathrm{a}\mathrm{set}\mathrm{of}\mathrm{twenty}\mathrm{six}\mathrm{consecutive}\mathrm{non}\mathrm{negative}\mathrm{integers} \\ \mathrm{represents}\mathrm{letter}\mathrm{of}\mathrm{alphabet}. \\ \mathrm{So}\mathrm{it}\mathrm{implies}\mathrm{that}, \\ \mathrm{MATHS}=\left(\mathrm{A}+12\right)+\mathrm{A}+\left(\mathrm{A}+19\right)+\left(\mathrm{A}+7\right)+\left(\mathrm{A}+18\right) \\ =\mathbf{5}\mathbf{A}\mathbf{+}\mathbf{56} \\ \mathrm{Take}\mathrm{A}=0\mathrm{and}\mathrm{A}=194 \\ 5\left(0\right)+56=56 \\ 5\left(194\right)+56=1026 \\ \mathrm{So}\mathrm{the}\mathrm{possible}\mathrm{forms}\mathrm{for}\mathrm{integral}\mathrm{A}\mathrm{are}\mathrm{only} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{56}\mathbf{}\mathbf{}\mathbf{and}\mathbf{}\mathbf{1026} \\ \mathbf{So}\mathbf{}\mathbf{we}\mathbf{}\mathbf{can}\mathbf{}\mathbf{have}\mathbf{}\mathbf{base}\mathbf{}\mathbf{3}\mathbf{}\mathbf{with}\mathbf{}\mathbf{A}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\mathbf{base}\mathbf{}\mathbf{8}\mathbf{}\mathbf{with}\mathbf{}\mathbf{A}\mathbf{=}\mathbf{194} \end{gathered}$$