Question

The product of three successive natural numbers, the middle number being the square of a natural number, is divisible by 60.
(A) True
(B) False

Answer 1

Hint: Natural numbers are those that are used for counting and ordering in mathematics. Words used for numbering are called "cardinal numbers," and words used for ordering are called "ordinal numbers" in modern mathematical terminology. Consecutive numbers are those that match each other in descending order from smallest to highest. For example, the numbers 1, 2, 3, 4, 5, 6, and so on are all consecutive.

Complete answer:
The other two numbers are (x-1) and (x+1) since the middle number of the three consecutive numbers is x.
The product of three successive natural numbers, the middle number being the square of a natural number, is divisible by 60
We have made a decision based on the given circumstances.
\[{x^2} = \left[ {{{(x + 1)}^2} - {{(x - 1)}^2}} \right] + 60\]
\[{x^2} = \left( {{x^2} + 2x + 1} \right) - \left( {{x^2} - 2x + 1} \right) + 60 = 4x + 60\]
Factorisation or factoring is defined in mathematics as the breaking or decomposition of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, which when multiplied together yield the original number.
\[{x^2} - 4x - 60 = 0\]
\[\left( {x - 10} \right)\left( {x + 6} \right) = 0\]
x = 10 or −6
Since x is a natural number, we get x=10
Hence the three numbers are 9, 10, 11.

Note:
Factorisation or factoring is defined in mathematics as the breaking or decomposition of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, which when multiplied together yield the original number.