Question

When \(13511\), \(13903\) and \(14589\) are divided by the greatest number \(n\), the remainder in each case is \(m\). The value of \((n+m)\) is:

• A. \(183\)
• B. \(182\)
• C. \(181\)
• D. \(179\)

Hint:

If several numbers leave the same remainder on division by \(n\), subtract the remainder first or take pairwise differences; the greatest such \(n\) is the \(\gcd\) of those differences.

Answer 1

The Correct Option is A

Step 1: Use the equal-remainder property.
If three numbers give the same remainder on division by \(n\), then their pairwise differences are multiples of \(n\).
Compute differences: \(13903-13511=392\), \quad \(14589-13903=686\), \quad \(14589-13511=1078\).
Step 2: Find the greatest such \(n\) as the GCD of the differences.
\(\gcd(392,686)=\gcd(392,686-392)=\gcd(392,294)=\gcd(294,98)=98\).
Also \(1078=98\times 11\Rightarrow 98\) divides all three differences.
Thus \(n=98\) (the greatest number).
Step 3: Find the common remainder \(m\).
Divide any of the numbers by \(98\):
\(13511=98\times 137 + 85 \Rightarrow m=85\).
(Check: \(13903=98\times 141+85\), \(14589=98\times 148+85\) — same remainder.)
Step 4: Compute \(n+m\).
\(n+m=98+85=183\). \[ \boxed{183} \]