Jacob guessed a natural number greater than 99 but less than 1000. The sum of the first and last digits of this number is 1, and the product of the first and second digits is 7. What number did Jacob guess?

Jacob guessed a natural number greater than 99 but less than 1000. The sum of the first and last digits of this number is 1, and the product of the first and second digits is 7. What number did Jacob guess?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

If the number is greater than 99 but less than 1000, then it is a three-digit number. Then let our number be abc = 100a + 10b + c.
The first digit of a three-digit number cannot be zero, so $a\ne 0$
The sum of the first and last digits of this number is 1.
The number looks like $abc⇒a+c=1$.
The sum of two non-negative numbers can be equal to one only if one of the numbers is equal to zero.
$a\ne 0$ and $a+c=1⇒c=0$
Substitute c=0 and find a.
$a+c=1⇒a=1-c=1-0=1$
The product of the first and second digits is 7.
The number looks like $abc⇒a\ast b=7$
Substitute a=1 and find b:
$a\ast b=7⇒b=7/a=7/1=7$.
a=1, b=7, c=0. Let's check the fulfillment of the conditions:
$1\right)99<170<999⇒99
$2\right)1+0=1⇒a+c=1$
$3\right)1\ast 7=7⇒ab=7$
Accordingly, the desired number is 170.