Question

If $R$ is an integer between $1$ and $9$, $P - R = 2370$, what is the value of $R$?

\(\textbf{Statements:}\)
I. $P$ is divisible by $4$. 
II. $P$ is divisible by $9$. 

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statements I and II are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

For data sufficiency with divisibility, reduce the known part modulo $m$ and isolate the unknown's residue class. Check if the range constraints yield a unique value.

Answer 1

The Correct Option is B

Step 1: Express $P$ in terms of $R$. 
Given $P - R = 2370 \Rightarrow P = 2370 + R$, with $R \in \{1,2,\dots,9\}$. 

Step 2: Analyze Statement I (divisible by 4). 
$2370 \equiv 2 \pmod{4}$, so $P \equiv 2 + R \pmod{4}$. 
For $P \equiv 0 \pmod{4}$, we need $R \equiv 2 \pmod{4} \Rightarrow R \in \{2,6\}$. 
Not unique $\Rightarrow$ I alone is insufficient. 

Step 3: Analyze Statement II (divisible by 9). 
$2370 \equiv 3 \pmod{9}$, so $P \equiv 3 + R \pmod{9}$. 
For $P \equiv 0 \pmod{9}$, need $R \equiv 6 \pmod{9}$. With $1\le R\le 9$, this forces $R=6$. 
Unique $\Rightarrow$ II alone is sufficient. \[ \boxed{\text{Answer (b): Statement II alone is sufficient.}} \]

Answer 2

This can also be settled by directly listing the nine possible values of \(P=2370+R\) for \(R=1,\dots,9\) and checking divisibility by 4 and by 9 in each case, rather than working through modular congruences.

Listing the values. For \(R=1,\dots,9\), \(P\) takes the values \(2371,2372,2373,2374,2375,2376,2377,2378,2379\).

Checking Statement I (divisible by 4). Only \(2372\) (when \(R=2\)) and \(2376\) (when \(R=6\)) are divisible by 4 among these nine values — two possibilities, so \(R\) is not pinned down uniquely.

Checking Statement II (divisible by 9). Since \(2370=9\times263+3\), we need \(3+R\) divisible by 9. Among \(R=1,\dots,9\), only \(R=6\) makes \(3+R=9\) divisible by 9, i.e. only \(P=2376\) is divisible by 9 — a single, unique possibility.

1. Option "Statement I alone": Insufficient — \(R=2\) and \(R=6\) both make \(P\) divisible by 4, so \(R\) can't be pinned down from this alone.
2. Option "Statement II alone": Sufficient — only \(R=6\) makes \(P\) divisible by 9, giving a unique value.
3. Option "Both statements needed": Not required — Statement II alone already fixes \(R=6\) uniquely, without needing Statement I at all.
4. Option "Cannot be answered": Incorrect — Statement II alone does determine \(R\) uniquely.

So the correct answer is the question can be answered with the help of statement II alone.