Let the number of columns be x.
x is the largest number, which should divide both 104 and 96
104 = 96 x 1 + 8 1
96 = 8 x 12 + 0
∴  HCF of 104 and 96 is 8
Hence, 8 columns are required.

Length = 8 m 50 cm = 850 cm
breadth = 6 m 25 cm = 625 cm
height = 4 m 75 cm = 475 cm
length of the longest rod is equal to

HCF (625, 850) = 25
∵ 25 divides 475
∴ HCF(62, 850, 475)=25

(15)ⁿ can end with the digit 0 only if (15)" is divisible by 2 and 5.
But prime factors of (15)ⁿ are 3ⁿ x 5ⁿ 
By Fundamental theorem of Arithmetic, there is no natural number II for which (15)ⁿ ends with the digit zero.

No.
15 does not divide 175.
LCM of two numbers should be exactly divisible by their HCF.
∴ Two numbers cannot have their HCF as 15 and LCM as 175

\(3\times 12\times 101+4=4(3\times 3\times 101+1)\)
 = 4(909+1)
 = 4(910)
= a composite number
[∵  Product of more than two factors]

∴ Composite number, x = 6762

y = 5 x 13 = 65
and x = 3 x 195 = 585

Given HCF (306, 1,314) =18
LCM (306, 1,314) = ?
Let  a = 306
b = 1,314
We know that
a x b = LCM (a, b) x HCF (a, b)
⇒ 306 x 1,314 = LCM (a, b) x 18
\(\Rightarrow \quad LCM(a,b)=\frac { 306\times 1,314 }{ 18 } \)
∴ LCM (306, 1,314) = 22,338

HCF of K
k.2
k.3
k.2²
k.5 is k

13; x = - 2, y =1

19000

36

Let a = n² - 1, where n = 1, 3, 5, ...
At n = 1, then a = (1)² - 1 = 1 - 1 = 0
At n = 3, then a = (3)² - 1 = 9 - 1 = 8
At n = 5, then a = (5)² - 1 = 25 - 1 = 24
which is divisible by 8.
Hence, n is an odd integer.

Here, 327.7081 is terminating. So, it represents a rational number.
Thus, \(327.7081=\frac { 3277081 }{ 10000 } =\frac { p }{ q } \)
Here, q = 10⁴ = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5
= 2⁴ x 5⁴ = (2 x 5)⁴
So, the prime factors of q are 2 and 5.

\(\because \) smallest odd composite number = 9
and smallest odd prime number = 3.
\(\therefore \) HCF of 9 and 3 = 3
and LCM of 9 and 3 = 9
Now, if an odd number p divides q², then p is one of the factors of q², i.e. q² = pm, for some integer m.    .... (i) 
Now, q³ = q² . q \(\Rightarrow \) q³ = pm . q   [from Eq.(i)]
\(\Rightarrow \)  q³ = p (mq)
\(\Rightarrow \)  p is a factor of q³ also \(\Rightarrow \)  p divides q³