Let a, b, c, d, e be natural numbers in an ar | vedclass

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Question

(b)

We have,

$a, b, c, d, e$ are natural number and in AP.

Let $D$ is common difference of $AP$.

$	herefore$ Let $c=C$

$a =C-2 D$
$b

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(b)

We have,

$a, b, c, d, e$ are natural number and in AP.

Let $D$ is common difference of $AP$.

$	herefore$ Let $c=C$

$a =C-2 D$
$b =C-D$

$d =C+D$

$e =C+2 D$

$a+b+c+d+e=5 C$

and $b+c+d=3 C$

Given, $a+b+c+d+e$ is a cube of number

$	herefore 5 C=\lambda^3$

and $b+c+d$ is a square of number

$	herefore 3 C=u^2$

From Eqs.$(i)$ and $(ii)$, we get

$\frac{\lambda^3}{5}=\frac{u^2}{3}$

$\lambda^3$ and $u^2$ is a multiple of 15

$	herefore$ Smallest possible value of $\lambda=15$ and $u=45$

$	herefore \quad c=\frac{u^2}{3}=\frac{(45)^2}{3}=675$

$	herefore$ Number of digits $=3$

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is

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