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find the smallest number which when increased by 17 is exactly divisible by both 520 and 468

The smallest number is 4663.

Find the smallest number which when increased by 17 is exactly divisible

by both 520 and 468

Quick Scoop

We want a number xxx such that when you add 17 to it, the result is divisible by both 520 and 468.

That means:

  • x+17x+17x+17 must be a common multiple of 520 and 468.
  • To make x+17x+17x+17 as small as possible, we choose the least common multiple (LCM) of 520 and 468.

So:

  • LCM(520,468)=4680(520,468)=4680(520,468)=4680.
  • x+17=4680⇒x=4680−17=4663x+17=4680\Rightarrow x=4680-17=4663x+17=4680⇒x=4680−17=4663.

✅ Answer: 4663 is the smallest number which when increased by 17 is exactly divisible by both 520 and 468.

Step-by-step breakdown

1. Set up the condition

Let the required number be xxx.

We are told:

  • x+17x+17x+17 is divisible by 520.
  • x+17x+17x+17 is also divisible by 468.

So x+17x+17x+17 is a common multiple of 520 and 468.

Since we want the smallest such xxx, x+17x+17x+17 should be the least common multiple of 520 and 468.

2. Use LCM idea

From standard math practice on such questions:

  • LCM(520,468)=4680(520,468)=4680(520,468)=4680.

Now:

  • x+17=4680x+17=4680x+17=4680.
  • So x=4680−17=4663x=4680-17=4663x=4680−17=4663.

3. Quick check

You can verify:

  • 4663+17=46804663+17=46804663+17=4680.
  • 4680÷520=94680÷520=94680÷520=9 (an integer).
  • 4680÷468=104680÷468=104680÷468=10 (an integer).

So 4680 is divisible by both 520 and 468, and 4663 is exactly 17 less than that, matching the condition.

Mini facts table (for clarity)

html

<table>
  <tr>
    <th>Item</th>
    <th>Value</th>
  </tr>
  <tr>
    <td>Numbers given</td>
    <td>520 and 468 [web:3][web:5][web:7][web:9]</td>
  </tr>
  <tr>
    <td>LCM(520, 468)</td>
    <td>4680 [web:3][web:5][web:7][web:9]</td>
  </tr>
  <tr>
    <td>Condition</td>
    <td>x + 17 is divisible by 520 and 468 [web:3][web:9]</td>
  </tr>
  <tr>
    <td>Equation</td>
    <td>x + 17 = 4680 [web:3][web:9]</td>
  </tr>
  <tr>
    <td>Smallest number x</td>
    <td>4663 [web:3][web:5][web:7][web:9]</td>
  </tr>
</table>

TL;DR

  • Take the LCM of 520 and 468 → 4680.
  • Subtract 17 → 4680−17=46634680-17=46634680−17=4663.
  • So the required smallest number is 4663.

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