Question

Let a, b, c be distinct digits. Consider a two digit number ‘ab’ and a three digit number ‘ccb’, both defined under the usual decimal number system. If $(ab)^2$ = ccb and ccb < 300, then the value of b is

• A. 0
• B. 1
• C. 2
• D. 5
• E. 6

Answer 1

The Correct Option is D

Let's break down the problem step by step to find the value of \( b \).

We are given:

• \( ab \) is a two-digit number formed by the digits \( a \) and \( b \). 
• \( ccb \) is a three-digit number formed by the digits \( c \), \( c \), and \( b \).
• The condition \( (ab)^2 = ccb \).
• \( ccb < 300 \).

Since \( ab \) is a two-digit number, \( ab = 10a + b \).

The number \( ccb \) is written in expanded form as \( 100c + 10c + b = 110c + b \).

Given that \( (ab)^2 = ccb \), we have the equation:

\((10a + b)^2 = 110c + b\)

Since \( ccb < 300 \), it implies:

• The number can be 100 to 299, so \( c \) can be at most 2 (as \( 110 \times 3 = 330 \) which is greater than 300).
• This means \( c \) can be 1 or 2.

Case Analysis:

1. When \( c = 1 \):
   • The number becomes \( ccb = 110 \times 1 + b = 110 + b \).
   • Then, \((10a + b)^2 = 110 + b\).
   • Solving: \((10a + b)^2 = 110 + b\).
   • Simplifying, \((10a + b)^2 - b = 110\).
   • Testing small values for \( a \) and corresponding \( b \).
2. When \( c = 2 \):
   • The number becomes \( ccb = 110 \times 2 + b = 220 + b \).
   • Then, \((10a + b)^2 = 220 + b\).
   • Simplifying, \((10a + b)^2 - b = 220\).
   • We find \((ab)\) such that \((ab)^2 = 245\). Run trials to find values of \( a \) and \( b \).

Let's compute specific values:

• If \( c = 1 \): \( (ab)^2 - b = 110 \) leads us by trials towards \( ab = 11\) which gives \( 121 \) not fitting.
• If \( c = 2 \): We compute to find a result fitting \( (ab)^2 = 245 \) leading us by trials towards \((ab) = 15\).

Using the correct \( ab = 15 \), we confirm \( 15^2 = 225 \), matching the expanded form \( 220 + 5 \).

Conclusion: The correct digit for \( b \) ensuring the number ccb satisfies the original conditions is:

The value of \( b \) is 5.