Question

A natural number is called a perfect square, if it is the square of some other natural number.

Note: Perfect squares are not going to end in 2, 3, 7 or 8 and they won't be of the form in $4n+2orin+3.$

Let n be a product of four consecutive positive integers then which answer is not true.

• A. $n+1$ is always a perfect square
• B. n is never a perfect square
• C. n is always divisible by 24
• D. $n+12$ is always a perfect square

Answer 1

Answer: (B), (C)

The correct options are
B n is never a perfect square
C n is always divisible by 24

$Letn=a(a+1)(a+2)(a+3)aspertheproblem.Case1-Ifaiseventhena=2pwhenpisaninteger.\therefore n=2p(2p+1)(2p+2)(2p+3)=4p(2p+1)(2p+1)(2p+3)Firsttermoftheproduct=16p^4\Rightarrow asquarenumber\left(1\right)Lasttermoftheproduct=1\times 2\times 3=6\Rightarrow notasquarenumber\left(2\right)Alsoweknowthatifakisintheform{k}^r=(l+m{)}^{r}and{k}^{r}isasquarenumber,theexpansionof(l+m{)}^{r}musthavefirstterm{l}^{r}andlastterm{m}^{r}assquarenumbers........\left(3\right)Case2-aisodd=2p+1thenn=(2p+1)(2p+2)(2p+3)(2p+4)=4(2p+1)(2p+1)(2p+3)(p+2)........\left(4\right)Firsttermoftheproduct=16p^4\Rightarrow asquarenumber\left(5\right)Lasttermoftheproduct=2\times 3\times 4=24\Rightarrow notasquarenumber......\left(6\right)Fromtheaboveresultsi.e1,2,3,4,5and6weinvestigatetheoptions.OptionA⟶Inn+1lastterm=6+1=7whenaisevenandlastterm=24+1=25=5^{2}whenaisodd.Son+1mayormaynotbeasquarenumber.OptionAisnotcorrect.OptionB⟶aeven\Rightarrow lastterm6aodd\Rightarrow lastterm24.Both6and24arenotsquarenumber.\therefore nisneveraperfectsquare.\therefore OptionBiscorrect.OptionC⟶Whenaisevenandtakesvalue2,4,6etcwehaven=120,840,3024,.................respectively.=24\times 5,24\times 35,24\times 126,.............\therefore nisdivisibleby24whenaisoddandtakesvalue1,3,5etcwehaven=24,360,1680-----respectively=24,24\times 15,24\times \mathrm{70...........}respectively\therefore nisalwaysdivisibleby24OptionCiscorrect.Alsoeverysetof4consecutivenumbercontains2,3,4asfactors.Soanyfourconsecutivenumberisdivisibleby24.OptionD⟶n+12\Rightarrow lasttermis12+6=18\Rightarrow notasquarenumberand12+24=36\Rightarrow asquarenumber.\therefore n+12mayormaynotbeasquarenumber.OptionDisnotcorrect.Answer-BandC$