To find the smallest number which, when increased by 17, is exactly divisible by both 520 and 468, we need to follow these steps:
### Step 1: Calculate the Least Common Multiple (LCM)
First, we need to calculate the Least Common Multiple (LCM) of 520 and 468.
1. Prime Factorization:
- \(\( 520 = 2^3 \times 5 \times 13 \)\)
- \(\( 468 = 2^2 \times 3 \times 3 \times 13 \)\) or \(\( 468 = 2^2 \times 3^2 \times 13 \)\)
2. LCM Calculation:
- LCM is found by taking the highest power of each prime factor appearing in the factorizations of both numbers.
- Highest power of 2: \(\( 2^3 \)\) (from 520)
- Highest power of 3: \(\( 3^2 \)\) (from 468)
- Highest power of 5: \(\( 5^1 \)\) (from 520)
- Highest power of 13: \(\( 13^1 \)\) (common in both)
Combining these, the LCM of 520 and 468 is:
\(\[
\text{LCM} = 2^3 \times 3^2 \times 5 \times 13
\]\)
Calculating this:
\(\[
2^3 = 8
\]\)
\(\[
3^2 = 9
\]\)
\(\[
\text{Multiplying these together: }
8 \times 9 = 72
\]\)
\(\[
72 \times 5 = 360
\]\)
\(\[
360 \times 13 = 4680
\]\)
So, the LCM of 520 and 468 is 4680.
### Step 2: Find the Smallest Number
We already know that when the smallest number is increased by 17, it should be divisible by both 520 and 468. This means the smallest number + 17 must be a multiple of their LCM, 4680.
### Step 3: Derive the Smallest Number
To find the smallest number:
\(\[
\text{Smallest number} = \text{LCM} - 17
\]\)
\(\[
\text{Smallest number} = 4680 - 17
\]\)
\(\[
\text{Smallest number} = 4663
\]\)
Thus, the smallest number which, when increased by 17, is exactly divisible by both 520 and 468, is 4663.
