Step 1 Given
W.K.T.

Multiples of 2 as well as of \(\displaystyle{5}={L}{C}{M}\) os (2,5)

\(\displaystyle={10}\)

Step 2 Finding the sum

Multiples of 2 as well as of 5 between 1 and \(\displaystyle{500}={10},{20},{30},\ldots,{490}\)

\(\displaystyle{a}_{{{1}}}={10}\)

\(\displaystyle{a}_{{{2}}}={20}\)

\(\displaystyle{d}={a}_{{{2}}}-{a}_{{{2}}}\)

\(\displaystyle={20}-{10}\)

Let

The number of terms in this \(\displaystyle{A}{P}={n}\)

\(\displaystyle{a}_{{{n}}}={a}+{\left({n}-{1}\right)}{d}\)

\(\displaystyle{490}={10}+{\left({n}-{1}\right)}{10}\)

\(\displaystyle{480}={\left({n}-{1}\right)}{10}\)

\(\displaystyle{n}-{1}={48}\)

\(\displaystyle{n}={49}\)

Sum of an AP

\(\displaystyle{S}_{{{n}}}={\frac{{{n}}}{{{2}}}}{\left[{a}+{1}\right]}\)

\(\displaystyle={\frac{{{49}}}{{{2}}}}{\left[{10}+{490}\right]}\)

\(\displaystyle={49}\times{250}\)

\(\displaystyle={12250}\)

Sum of those integers between 1 and 500 which are multiples of 2 as well as of \(\displaystyle{5}={12250}\)

Multiples of 2 as well as of \(\displaystyle{5}={L}{C}{M}\) os (2,5)

\(\displaystyle={10}\)

Step 2 Finding the sum

Multiples of 2 as well as of 5 between 1 and \(\displaystyle{500}={10},{20},{30},\ldots,{490}\)

\(\displaystyle{a}_{{{1}}}={10}\)

\(\displaystyle{a}_{{{2}}}={20}\)

\(\displaystyle{d}={a}_{{{2}}}-{a}_{{{2}}}\)

\(\displaystyle={20}-{10}\)

Let

The number of terms in this \(\displaystyle{A}{P}={n}\)

\(\displaystyle{a}_{{{n}}}={a}+{\left({n}-{1}\right)}{d}\)

\(\displaystyle{490}={10}+{\left({n}-{1}\right)}{10}\)

\(\displaystyle{480}={\left({n}-{1}\right)}{10}\)

\(\displaystyle{n}-{1}={48}\)

\(\displaystyle{n}={49}\)

Sum of an AP

\(\displaystyle{S}_{{{n}}}={\frac{{{n}}}{{{2}}}}{\left[{a}+{1}\right]}\)

\(\displaystyle={\frac{{{49}}}{{{2}}}}{\left[{10}+{490}\right]}\)

\(\displaystyle={49}\times{250}\)

\(\displaystyle={12250}\)

Sum of those integers between 1 and 500 which are multiples of 2 as well as of \(\displaystyle{5}={12250}\)