Question

Some equations are solved on a certain basis. Find \(13 \times 13 = \,?\) given: \(5 \times 8 = 28,\; 3 \times 7 = 12,\; 8 \times 6 = 35.\)

• A. 169
• B. 130
• C. 140
• D. 144

Hint:

Test simple transforms like \((a-1)(b-1)\) or \((a+1)(b+1)\) on all examples to uncover the rule.

Answer 1

The Correct Option is D

Rule: \(a \times b \;\mapsto\; (a-1)(b-1)\). \[ 5\times 8 \Rightarrow 4\cdot 7=28,\quad 3\times 7 \Rightarrow 2\cdot 6=12,\quad 8\times 6 \Rightarrow 7\cdot 5=35. \] Therefore, \[ 13\times 13 \Rightarrow 12\cdot 12=144. \] \[ \boxed{144} \]

Answer 2

Rewrite the coding rule algebraically as \(ab-(a+b)+1\), which is the same as \((a-1)(b-1)\), and check it against the three given examples before testing each option for \(13\times13\).

1. Rule check: \(5\times8\Rightarrow40-(5+8)+1=28\); \(3\times7\Rightarrow21-(3+7)+1=12\); \(8\times6\Rightarrow48-(8+6)+1=35\) — the rule holds in every case.
2. Option (a) 169: Applying the rule to \(13\times13\) gives \(169-(13+13)+1=169-26+1=144\), not \(169\) — rejected.
3. Option (b) 130: Does not equal the rule's result of \(144\) — rejected.
4. Option (c) 140: Does not equal \(144\) — rejected.
5. Option (d) 144: Matches \(169-26+1=144\) exactly — confirmed.

The rule consistently gives \(144\) for \(13\times13\), so option (d) is correct.

the correct answer is 144.