Here, we have 34
34 = 31 + 3 or 29 + 5

1, 2, 3, 4, 5, 6,10,12,15,20,30,60

We have, 27
27 = 1x 27; 27 = 3 x 9
\(\therefore\)Factors of 27 are 1, 3, 9 and 27.

The required number is the HCF of (753 - 3) and (1054 - 4) i.e. 750 and 1050.

\(\therefore\)The HCF of 750 and 1050 is 150.
Hence, the required number is 150.

49 = 3 + 5 + 41

We have, 1258
(i) Divisibility by 2
\(\because\)Unit digit of number = 8, so 1258 is divisible by 2.
(ii) Divisibility by 3
Sum of digits of given number = 1+ 2 + 5 + 8 = 16 is not divisible by 3, so 1258 is not divisible by 3. Now, we see that 1258 is divisible by 2 but not divisible by 3.
Hence, it is not divisible by 6.

Let the required unknown digit be x.
Then, the number becomes
\(\begin{matrix} 8 \\ \downarrow \\ E \end{matrix}\begin{matrix} x \\ \downarrow \\ O \end{matrix}\begin{matrix} 9 \\ \downarrow \\ E \end{matrix}\begin{matrix} 4 \\ \downarrow \\ O \end{matrix}\begin{matrix} 8 \\ \downarrow \\ E \end{matrix}\begin{matrix} 4 \\ \downarrow \\ O \end{matrix}\)
Sum of digits at odd places from right = 4 + 4 + x = 8+x
Sum of digits at even places from right = 8 + 9 + 8= 25
\(\because\) Number is divisible by 11.
\(\therefore\) Difference of digits will be 0 or 11.
\(\Rightarrow\)25 - (8 + x) = 0 or 11
\(\Rightarrow\)25 - 8 - x = 0 or 11 \(\Rightarrow\) 17 - x = 0 or 11
Taking difference 0, 17-x=0 \(\Rightarrow\) x=17+0 \(\Rightarrow\) x=17 [but 17 is not a single digit number, so it is not possible]
Taking difference 11, 17 - x = 11 \(\Rightarrow\) x = 17 - 11= 6
So, required digit to write in the blank space is 6.