In a four-digit number, the sum of the digits of the thousands and tens is equal to $4$ . The sum of the digits of the hundreds and the units is $15$ , and the digit of the units exceeds by $7$ the digit of the thousands. Among all the numbers satisfying these conditions, find the number the sum of the product of whose digit of the thousands by the digit of the units and the product of the digit of the hundreds by that of the tens assumes the least value.

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Solution

let the number be $a b c d$ .
According to the statement:
$a + c = 4$
$b + d = 15$
$d - a = 7$

From the equations, following conclusions can be drawn:
$a$ can be $1, 2, 3, 4$ from $a + c = 4$ , but a cannot be $3$ and $4$ from $d - a = 7$ as $d$ exceeds $9$ , which is not possible.

For $a = 1$ , the number is $1738$ and for $a = 2$ the number is $2629$
Hence, the required number is $1738$ as we were supposed to find the least value.

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