What is the smallest integer greater than 5 that leaves a remainder of 5 when divided by any of the integers 6,8, and 10?
Looks like LCM(6,8,10)+5 = 125
485
To find the smallest integer greater than 5 that leaves a remainder of 5 when divided by 6, 8, and 10, we can use the concept of the least common multiple (LCM) of these three numbers.
The LCM of 6, 8, and 10 is the smallest number that is divisible by all three numbers. To find the LCM, we can start by finding the prime factorization of each number:
6 = 2 * 3
8 = 2 * 2 * 2
10 = 2 * 5
Now, we consider the highest power of each prime factor that appears in any of the numbers:
2^3 * 3 * 5 = 120
Therefore, the LCM of 6, 8, and 10 is 120.
To find the smallest integer greater than 5 that leaves a remainder of 5 when divided by any of these numbers, we can start with 120 and add 5 until we find a suitable number:
120 + 5 = 125 (divisible by 6? No. divisible by 8? No. divisible by 10? No.)
125 + 5 = 130 (divisible by 6? No. divisible by 8? No. divisible by 10? No.)
...
130 is not divisible by any of the numbers 6, 8, or 10. Let's test 131:
131 + 5 = 136 (divisible by 6? No. divisible by 8? No. divisible by 10? No.)
...
136 is not divisible by any of the numbers 6, 8, or 10. Let's test 137:
137 + 5 = 142 (divisible by 6? No. divisible by 8? No. divisible by 10? No.)
...
142 is not divisible by any of the numbers 6, 8, or 10. Let's test 143:
143 + 5 = 148 (divisible by 6? No. divisible by 8? No. divisible by 10? No.)
...
148 is not divisible by any of the numbers 6, 8, or 10. Let's test 149:
149 + 5 = 154 (divisible by 6? No. divisible by 8? No. divisible by 10? No.)
...
154 is not divisible by any of the numbers 6, 8, or 10. Let's test 155:
155 + 5 = 160 (divisible by 6? No. divisible by 8? Yes. divisible by 10? No.)
Therefore, the smallest integer greater than 5 that leaves a remainder of 5 when divided by 6, 8, and 10 is 160.