Try this beautiful problem from Algebra based on Integer
The equation \(166 \times 56 = 8590\) is valid in some base \(b \geq10\) (that is \(1,6,5,8,9,0\)) are digits in base b
in the above equation). Find the sum of all possible values of \(b \geq10 \)satisfying the equation.
Algebra
Integer
base
Answer:$12$
PRMO-2018, Problem 4
Pre College Mathematics
Arrange the given number
Can you now finish the problem ..........
Arrange the equation \(166 \times 56 = 8590\) in base b where\(b\geq 10\)
Can you finish the problem........
Given condition is \(b \geq10\)
Now
\(166 = b^2 + 6b + 6\)
\(56 = 5b + 6\)
\(98590 = 8b^3 + 5b^2 + 9b\)
According to the problem,\((b^2 + 6b + 6) (5b + 6) = 8b^3 + 5b^2 + 9b\)
\(\Rightarrow 5b^3 + 36b^2 + 66b + 36 = 8b^3 + 5b^2 + 9b\)
\(\Rightarrow 3b^3 – 31b^2 – 57b – 36 = 0\)
\(\Rightarrow (b – 12) (3b^2 + 5b + 3) = 0\)
\(\Rightarrow b = 12\)
We have only one b which is 12
so the required sum is 12
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.