Select the option that is related to the third number in the same way as the second number is related to the first number and the sixth number is related to the fifth number.
32 : 35 :: 34 : ? :: 36 : 243

Let's analyze the relationship between the pairs of numbers:
First pair: 32:35

Product of digits of 32=3×2=6

Sum of digits of 32=3+2=5

Let's check power relations: (3−2)=1, (3+2)=5.

Let's try: 3
2
+2
5
=9+32=41 (Not 35).

Let's try: Base
exponent
combinations:
For 36:243: We know 243=3
5
. Notice that the first digit is 3. The exponent 5 can be obtained from 6−1.
Let's check the structure: First digit
(Second digit−1)
.

For 36: 3
(6−1)
=3
5
=243. This works perfectly!

For 32: 3
(2−1)
=3
1
=3

=35. Let's modify the rule.

Alternative pattern:
Let's check (
2
Number
​
) or similar:

For 36: (
4
36
​
)
x
? No.

Let's check the digits again:
For 36: 3×6=18→18
2
=324.
What about 3
5
=243? The digits are 3 and 6. 36→3
(6−1)
worked for the second one, why didn't it work for 32? Let's check 32→ maybe it is 3
(2+1)
=27? No, it's 35.
Wait! Let's check:

32→(3+2)×7=35?

36→(3+6)×27=243? Notice that 7=2
3
−1 and 27=3
3
? No.
Let's look at the multiplier:

For 32: 5×7=35 (where 7=3
2
−2?)

For 36: 9×27=243 (where 27=3
3
)
Notice the multipliers: 7 and 27.
Let's look at the second digit: For 32, second digit is 2→2
3
−1=7. For 36, second digit is 6→ doesn't fit 27.
Wait, look at 7 and 27:

35=32+3

243=36×6.75
Let's reconsider the digit exponents:

32→3
3
+2
3
=27+8=35. Wow! Sum of cubes of the digits!
Let's verify this rule for 36:

36→3
3
+6
3
=27+216=243. Yes! It fits perfectly!

Now, apply this rule to 34:

34→3
3
+4
3
=27+64=91.