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.
