Problem Solving

The average (arithmetic mean) of 4 different positive integers is 125 and the largest of these integers is 150, what is the least possible value of the smallest of the 4 integers?

A 1
B 2
C 12
D 53
E 100

Let the 4 different positive integers be a, b, c, and d
Largest integer = 150
Let the largest integer = d
d = 150


$$\frac{a+b+c+150}{4}=125$$
$$a+b+c+150=125\cdot4$$
$$a+b+c=500-150$$
$$a+b+c=350$$


To get the least possible value of the smallest of the 4 integers, we have to maximize the other 2 integers
Let least/smallest integer = a


Since largest integer = 150, the maximum possible value for other 2 integer= 150 - 1 and 150 - 2 respectively


Smallest integer => a + (150 - 2) + (150 - 1) = 350
=> a + 148 + 149 = 350
a + 297 = 350
a = 350 - 297
a = 53

Answer = D