Valentine Vinculum Part 2

We are given that A is able to figure out the entire 4-digit code. If the code isn't all the same digits, then there exist various permutations which would make it unable for him to figure it out. Hence, we can conclude that the code is of the form $\overline{ aaaa}$ .

Now, he knows the product is $a^ 4$ . Since he was previously unable to figure out what the code must be, we know that there is more than 1 way to factorize $a^4$ using only 4 digits.

Hence, we could have:
( $1^ 4$ has no other factorization)
$2^ 4$ with another factorization like $1 \times 2 \times 2 \times 4$
$3^ 4$ with the other factorization like $1 \times 3 \times 3 \times 9$
$4^ 4$ with the other factorization like $2 \times 4 \times 4 \times 8$
( $5^ 4$ has no other factorization)
$6^ 4$ has other factorizations like like $4 \times 4 \times 9 \times 9$
( $7^ 4$ has no other factorization)
( $8^ 4$ has no other factorization)
( $9^ 4$ has no other factorization)

We still have one remaining fact, which is that the square of the (product of the digits) starts with 1. We then check that $(2^4)^2 = 256, (3^4)^2 = 6562, (4^4) ^2 = 65356, (6^4)^2 = 1679616$ .

Thus, the only possibility is that the code is 6666, and the phone number is 1679616,

• code=area code
• code= $xyzt$
• we know code is 4 digit number
• $(code)$ ^2=1..(unknown numbers of digit)
• $(x.y.z.t)$ ^2 can be minimum 1000000 becuase $x.y.z.t$ can be minimum 1000
• $(x.y.z.t)$ ^2 can be maximum 1999396(because phone number contains 1 as left most digit) $x.y.z.t$ can be maximum 1414
• 1000≤ $x.y.z.t$ ≤1414 1000000≤phone number≤1999396
• Agnishom knows the [ $x.y.z.t$ ]'s result but he couldn't find out number however Agnishom's Love says if she tells Agnishom sum of digits Agnishom can figure out it exactly.
• the sentence above simply means all numbers of xyzt is same because even if one number is diffrent,it can be formed another number such as yzxt,tyzx,zytx.. (becuase Commutative property of multiplication) and it makes Agnishom to find girl's phone number impossible.
• 1^4=1 1<1000 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 2^4=16 16<1000 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 3^4=81 81<1000 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 4^4=256 256<1000 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 5^4=625 625<1000 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 6^4=1296 1000< $1296$ <1414 DEFINITELY SUPPORTS INEQUALITY
• 7^4=2401 2401>1414 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 8^4=4096 4096>1414 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• 9^4=6561 6561>1414 does not support inequality 1000≤ $x.y.z.t$ ≤1414
• [9.4.9.4 = 6.6.6.6 9+4+9+4=26 6+6+6+6=24 it is proof why sum is necessary]
• so the area code is $6666$