If 4 # 2 = 10 and 6 # 3 = 21, what is 8 # 4?

Correct — A.
Rule: a # b = (a + 1) × b, the same as a × b + b.

4 # 2 = 5 × 2 = 10, as given
6 # 3 = 7 × 3 = 21, as given
8 # 4 = 9 × 4 = 36 → option (a).

Why the others are wrong

• (b)
  40 — 40 is 8 × 4 + 8, adding the first number instead of the second. Tested on 4 # 2 that rule gives 8 + 4 = 12, not the given 10.
• (c)
  44 — 44 is 11 × 4, a rule of (a + 3) × b. Tested on 4 # 2 it gives 7 × 2 = 14, not the given 10.
• (d)
  48 — 48 is 8 × 4 + 16, adding b² instead of b. Tested on 4 # 2 it gives 8 + 4 = 12, not the given 10.

An operator defined by examples hides a formula. Find one formula that reproduces every given result, then apply it to the new pair.

Start from the product: 4 × 2 = 8 and 6 × 3 = 18. Each result sits exactly b above it (8 + 2 = 10, 18 + 3 = 21), so a # b = a × b + b.

A rule tested on one example is a guess. A rule that fits both is the working rule.

Key facts

• a × b + b = (a + 1) × b.
• 10, 21 and 36 are the 4th, 6th and 8th triangular numbers.
• Test a candidate rule on every given example before using it.

Common traps

• Stopping at a rule that fits 4 # 2 alone: 4² − 3 × 2 = 10, but 6² − 3 × 3 = 27, not 21.
• Adding the first number to the product instead of the second, which gives 40.

No directly related past PYQ was found.