Question

Prashant forgot his five-digit ATM pin. However he remembers that the ATM PIN has a cube of a number in the 3rd place and prime numbers in the first and the last places. The even prime number is in the 1st place. He also remembers that the digit in the 3rd place is equal to the product of the digits in the 1st and the 2nd places and the sum of the digits in the 1st and the 2nd places is equal to the digit in the 4th place. The digit in the last place is half the digit in the 4th place. What is his ATM PIN?

• A. 24863
• B. 24683
• C. 26483
• D. 23468

Hint:

Turn each sentence into an equation/constraint by position (product, sum, half, parity, prime). Try small cases quickly—often only one case survives all constraints.

Answer 1

The Correct Option is A

Let the digits be \(a\,b\,c\,d\,e\).
Given: \(a\) and \(e\) are prime; the even prime is in the 1\textsuperscript{st} place \(\Rightarrow a=2\).
The 3\textsuperscript{rd} digit is a cube digit \(\Rightarrow c\in\{0,1,8\}\).
Also \(c=a\cdot b \Rightarrow c=2b\).
Thus \(2b\in\{0,1,8\}\Rightarrow 2b\in\{0,8\}\Rightarrow b\in\{0,4\}\).
Case 1: \(b=0\). Then \(c=0\), \(d=a+b=2\), \(e=d/2=1\). But \(e\) must be prime; \(1\) is not prime \(\Rightarrow\) reject.
Case 2: \(b=4\). Then \(c=2\times4=8\) (a cube digit ✓).
Next \(d=a+b=2+4=6\).
Finally \(e=d/2=3\) which is prime ✓.
Therefore the PIN is \(a\,b\,c\,d\,e=2\,4\,8\,6\,3\).
\(\boxed{24863}\)