The least multiple of 13 which when divided by 4, 5, 6, 7 leaves remai...
To find the least multiple of 13 that leaves a remainder of 3 when divided by 4, 5, 6, and 7, we can use the concept of the least common multiple (LCM).
Step 1: Find the LCM of 4, 5, 6, and 7.
The LCM of 4, 5, 6, and 7 can be found by listing their multiples and finding the smallest common multiple.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, ...
From the lists above, we can see that the smallest common multiple is 60. Therefore, the LCM of 4, 5, 6, and 7 is 60.
Step 2: Find the smallest multiple of 13 that leaves a remainder of 3 when divided by 60.
To find the smallest multiple of 13 that leaves a remainder of 3 when divided by 60, we can start by finding the first multiple of 13 that is greater than 60.
Multiples of 13: 13, 26, 39, 52, 65, 78, ...
From the list above, we can see that the first multiple of 13 greater than 60 is 65.
Now, we can subtract 60 from 65 to find the remainder: 65 - 60 = 5.
Since the remainder is not 3, we need to keep subtracting 60 from the multiple of 13 until we get a remainder of 3.
65 - 60 = 5 (remainder is not 3)
65 - 60 - 60 = 5 - 60 = -55 (remainder is not 3)
65 - 60 - 60 - 60 = 5 - 60 - 60 = -115 (remainder is not 3)
65 - 60 - 60 - 60 - 60 = 5 - 60 - 60 - 60 = -175 (remainder is not 3)
65 - 60 - 60 - 60 - 60 - 60 = 5 - 60 - 60 - 60 - 60 = -235 (remainder is not 3)
65 - 60 - 60 - 60 - 60 - 60 - 60 = 5 - 60 - 60 - 60 - 60 - 60 = -295 (remainder is not 3)
Finally, when we subtract 60 from 65 six times, we get a remainder of 3: 65 - 60 - 60 - 60 - 60 -