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Example 3:

Find the least number which, when divided by 8, 12, 20, and 36, leaves remainders 6, 10, 18, and 34 respectively.

Solution:

Here, the numbers are 8, 12, 20, and 36, and the respective remainders are 6, 10, 18, and 34.

The difference between each number and its respective remainder is equal to 2.

First, find the LCM of 8, 12, 20, and 36, which is 360.

The required number is [tex]\(360 - 2 = 358\)[/tex].

Sagot :

GinnyAnswer
Certainly! Let's walk through the problem step by step:

1. Given Information:
- We need to find the least number which, when divided by 8, 12, 20, and 36, leaves remainders of 6, 10, 18, and 34 respectively.

2. Understand the Remainders:
- For the number [tex]\( x \)[/tex] we are looking for:
- When [tex]\( x \)[/tex] is divided by 8, it leaves a remainder of 6.
- When [tex]\( x \)[/tex] is divided by 12, it leaves a remainder of 10.
- When [tex]\( x \)[/tex] is divided by 20, it leaves a remainder of 18.
- When [tex]\( x \)[/tex] is divided by 36, it leaves a remainder of 34.

3. Difference Observation:
- Notice the pattern in remainders:
- [tex]\( 8 - 6 = 2 \)[/tex]
- [tex]\( 12 - 10 = 2 \)[/tex]
- [tex]\( 20 - 18 = 2 \)[/tex]
- [tex]\( 36 - 34 = 2 \)[/tex]
- All these differences are equal to 2.

4. Adjusting the Problem:
- This tells us that the number [tex]\( x \)[/tex] when divided by each of 8, 12, 20, and 36 leaves a remainder that is 2 less than each divisor.
- We can reframe the problem as finding a number [tex]\( y \)[/tex]:
- [tex]\( y = x + 2 \)[/tex]
- This means [tex]\( y \)[/tex] is divisible exactly by 8, 12, 20, and 36.

5. Finding the LCM:
- To find [tex]\( y \)[/tex], it should be the Least Common Multiple (LCM) of 8, 12, 20, and 36.
- The LCM of 8, 12, 20, and 36 is 360.

6. Concluding the Problem:
- Now, [tex]\( x = y - 2 \)[/tex]
- Since [tex]\( y = 360 \)[/tex], [tex]\( x = 360 - 2 = 358 \)[/tex].

So, the least number which when divided by 8, 12, 20, and 36 leaves remainders 6, 10, 18, and 34 respectively is 358.
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