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Q.

State true or false.
The cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
 


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a

True

b

False 

answer is A.

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Detailed Solution

Given a statement the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Assuming a as the positive integer whose cube is to be calculated.
And let b be also a positive integer.
According to Euclid’s division lemma, the relation of a and b is
a=bq+r  .    Here, r = remainder and q = quotient.
Assuming the value of b = 3.
Possible remainders are 0,1,2.
Putting r=0   in the above equation,
a=3×q+0  
a=3×q  
Now, taking the cube on both sides,
= (3q) 3 =27 q 3 =9×3 q 3 =9m                 [m=3 q 3 ]  
Hence, it is of 9m form.
Putting r = 1 in equation, 
a=3×q+1  .
Now, taking the cube on both sides,
Question Image
Hence, it is of 9m + 1 form.
Putting r = 2 in the equation,
a=3×q+2  .
Now, taking the cube on both sides,
= (3q+2) 3  
= (3q) 3 +3× 3q 2 ×2+3×3q× 2 2 + 2 3               [As, a+b 3 = a 3 +3 a 2 b+3a b 2 + b 3 ]  
=27 q 3 +108 q 2 +36q+8 = 27 q 3 +108 q 2 +36q +8 =9× 3 q 3 +12 q 2 +4q +8 =9m+8                                            [m=3 q 3 +12 q 2 +4q]  
Hence, it is of 9m + 8 form.
Therefore, proving the given statement is true.
Hence, option 1 is correct.
 
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