Latest CAT Aptitude Question SOLUTION: The sum of the four consecutive two digit odd number when divided by 10 becomes a perfect square, which of the following can be one of these four numbers? a.21 b.25 c.41 d.67 e.73

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The sum of the four consecutive two digit odd number when divided by 10 becomes a perfect square, which of the following can be one of these four numbers?
a.21 b.25 c.41 d.67 e.73

Asked In CAT Devendra Marghade (12 years ago)

Unsolved Read Solution (11)

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Read Solution (11) :

chaitanya  (12 years ago)

Let the four 2-digit odd numbers be
n-3 n-1 n+1 n+3

Sum of the 4 numbers ==> 4n

acc to qn, when the sum is divided by 10
we get a perfect square...
perfect squares include==>1,4,9,16,25,36,49,.....

Possible values of 4n/10 ==> 4, 16,36...

If 4n/10=4
n=10
Hence, the corresponding nos are 7,9,11,13(all of which are NOT 2-digit nos)


If 4n/10=16
n=40
Hence the corresponding nos are 37, 39, 41, 43

If 4n/10=36
n=90
Hence the corresponding nos are 87, 89, 91, 93


The answer to the ques therefore is Option C

Is this solution Helpfull? Yes (33) | No (5)  

chaitanya  (12 years ago)

Let the no be : 10 X +Y , {10 X+Y+2}, {10 X+Y+4}, {10 X+Y+6}

Sum of 4 consecutive 4 odd integers will be 40 X+ 4Y +12.

As per the stem, sum divided by 10 will be a perfect square.

Let's list down the perfect squares : 1,4,9,16,25 ...

SO the sum will be one among 10,40,90,160,250..... { plz note I have multiplied the perfect sqaures with 10).

We know that Y can be of 1,3,5,7,9 ( as Y is a odd integer)

when X = 3, Y = 7 , the sum becomes 160.

So the odd integer is 37.

Remaining 3 integers will be 39,41,43.

We can see that 41 is one among the options pick C.

Is this solution Helpfull? Yes (11) | No (4)  

Vignesh  (12 years ago)

C. 41

Easy way.
We know that Odd digits are 1,3,5,7 and 9.

If sum of four consecutive odd numbers is divisible by 10 means,

Then the possible four digits which divisible by 10 should be 1,3,7,9.....
(i.e., 1+3+7+9 = 20 which divisible by 10)

Now by using option C. 47 , we form the numbers 37,39,41,43.

So, 41 is one of those numbers.

Is this solution Helpfull? Yes (10) | No (13)  

nvj  (8 years ago)

(X+(X+2)+(X+4)+(X+6))/10 ==perfect square(16)
X=37
then 39,41,43
ans:41

Is this solution Helpfull? Yes (5) | No (1)  

Chirag Ranjan Lahiri  (9 years ago)

We know that Odd digits are 1,3,5,7 and 9.

If sum of four consecutive odd numbers is divisible by 10 means,

Then the possible four digits which divisible by 10 should be 1,3,7,9.....
(i.e., 1+3+7+9 = 20 which divisible by 10)

Now by using option C. 47 , we form the numbers 37,39,41,43.

So, 41 is one of those numbers.

Is this solution Helpfull? Yes (4) | No (4)  

Anusha.C  (8 years ago)

assume numbers to be x-3, x-1, x+1, x+3. Sum of these numbers is 4x.

Now divide this by 10. This gives us 2x/5.

For what value of x is 2x/5 a perfect square? Start equating 2x/5 to perfect squares.
=> x=40 gives us 16 - a perfect square
This gives us 41 as the answer.

Is this solution Helpfull? Yes (2) | No   

raghuram dharmavarpu  (9 years ago)

these are the four odd integers 2x-3,2x-1,2x+1,2x+3
the sum of the above four numbers is 8x
for two digit number x min value is 7 and maximum value is 47
we know that 8x/10 is a perfect square
the perfect squares 1,4,9,16,25,36,49,64..
for x=20 we can get the answer as 8(20)/10 =16
16 is perfect square. put the x value in the above equation we can get the values 37,39,41,43

Is this solution Helpfull? Yes (1) | No (2)  

rahul  (8 years ago)

ANSWER ; C

Is this solution Helpfull? Yes | No (2)  

Rajesh  (7 years ago)

let the four consecutive nos x-3,x-1,x+1,x+3
sum of above nos is 4x
when we divide it by 10 (it must be perfect no)
we know 4*40=160
160/10=16
its perfect square
hence x=40
let four consecutive no are 37,39,41,43

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vinay  (7 years ago)

Let the numbers are 2a +1, 2a + 3, 2a + 5, 2a + 7 and their sum = 8a + 16
Given that if this sum is divided by 10, results in a perfect square.
So, (8a+16)/10 = k^2.
⇒a = (10k^2−16) /8
⇒a=[ 5(k^2)/4 ]−2
As k^2 is a perfect square and has to be divided by 4, only even numbers should be considered for k.
For k = 2, we get, a = 3, but 2a + 1 is not a two digit number.
For k = 4, a = 18 for which the given condition is satisfying.
So the numbers are 37, 39, 41, 43.
For k = 6, a = 43. The number are 87, 89, 91, 93.

Is this solution Helpfull? Yes | No (2)  

  (4 years ago)

Let the four 2 digit odd numbers be
x+1, x+3, x+5, x+7

Then x+1+x+3+x+5+x+7= 4x+16

according to the question, when the sum is divided by 10
we get a square.
squares include=1,4,9,16,25,36,49,.....

Possible values of (4x+16)/10 = 4, 16,36...

If (4x+16)/10=4
x=6
Hence, the corresponding odd numbers are 7,9,11,13(but 7 & 9 are not 2 digit numbers)


If (4x+16)/10=16
x=36
Hence the corresponding odd numbers are 37, 39, 41, 43

If (4x+16)/10=25
x=58.5
but this number is not integer number.

If (4x+16)/10=36
x=86
Hence the corresponding odd numbers are 87, 89, 91, 93

If (4x+16)/10=49
x=118.5
but this number is not integer number and also not 2 digit number.

so the answer is c.41

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