Question

# Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

Factors and multiples
Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

2021-08-03
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 $$\displaystyle{5}={12250}$$