Question

Find the sum of the smallest two numbers out of three consecutive numbers whose product is given by 210.
(a) 11
(b) 15
(c) 20
(d) 56

Answer 1

Hint: In this question, in order to determine the sum of the smallest two numbers out of the three consecutive integers, we have to first assume that the three consecutive integers are given by \[x\], \[x+1\] and \[x+2\]. Then using the given information that the product of three consecutive numbers \[x\], \[x+1\] and \[x+2\] is 210, we will form an equation in variable \[x\]. We will then factorize 210 and represent it as the product of three consecutive numbers if possible. Thus we can determine the value of \[x\], \[x+1\] and \[x+2\]. Consequently, we can then determine the sum of the smallest two numbers.

Complete step-by-step solution:
Let us suppose that the first number is given by \[x\].
Then the first consecutive number for \[x\] is given by \[x+1\].
And the second consecutive number for \[x\] is given by \[x+2\].
Now it is given that the product of three consecutive numbers \[x\], \[x+1\] and \[x+2\] is 210.
We will now form an equation using the above fact.
\[\left( x \right)\left( x+1 \right)\left( x+2 \right)=210................(\text{i})\]
Now in order to find the value of \[x\], we have to solve the above equation.
On factorising 210, we get
\[\begin{align}
  & 3\left| \!{\underline {\,
  210 \,}} \right. \\
 & 7\left| \!{\underline {\,
  70 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\,\left| \!{\underline {\,
  5 \,}} \right. \\
 & \,\,\,\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
That is
\[\begin{align}
  & 210=2\times 3\times 5\times 7 \\
 & =5\times 6\times 7....................(\text{ii})
\end{align}\]
Where 5, 6 and 7 are three consecutive natural numbers.
Now equating the value of 210 in equation (i) and (ii) where \[x\], \[x+1\] and \[x+2\] are three consecutive integers and 5, 6 and 7 are also three consecutive natural numbers, we get
\[\left( x \right)\left( x+1 \right)\left( x+2 \right)=5\times 6\times 7\]
Thus we get
\[x=5\]
\[x+1=5+1=6\]
\[x+2=5+2=7\]
Here, the smallest two numbers of of the three consecutive integers 5,6 and 7 are 5 and 6.
Therefore the sum of the smallest two numbers is given by
\[5+6=11\]
Hence option (a) is correct.

Note: In this problem, we can also form an equation in the variable \[x\] using the given information that the product of three consecutive numbers \[x\], \[x+1\] and \[x+2\] is 210. Using this we will get an equation \[\left( x \right)\left( x+1 \right)\left( x+2 \right)=210\]. We can solve this equation to find the value of \[x\] by solving the equation of three degrees and then find the three consecutive integers. solving the 3 degree equation will be a lengthier approach so we avoid using this method.