#### Eric Barnes

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Solving using factorisation method:
55 = 5 x 11
130 = 2 x 5 x 13
185 = 5 x 37
lcm = 2 x 5 x 11 x 13 x 37 = 52910

Three million seven

No.
A number is divisible by 3 if, and only if, the sum of its digits
is also divisible by 3.
The test can be repeated on the sum, so repeatedly sum the digits
…until a single digit remains; only if this digit is 3, 6, 9 will
the original number be divisible by 3.
For 988876:
988876 â†’ 9 + 8 + 8 + 8 + 7 + 6 = 46
46â†’ 4 + 6 = 10
10 â†’ 1 + 0 = 1
1 is none of {3, 6, 9}, thus 988876 is not divisible by 3
The remainder when 988876 is divided by 3 is 1. (MORE)

Eighty-five billion

16 feet

Out of that list, just 5.
To be divisible by 2 the last digit must be even, ie one of {0, 2,
4, 6, 8}
The last digit is 5 which is not even (it is odd), so 745 i…s not
divisible by 2
To be divisible by 3 the sum of the digits must also be divisible
by 3; if the summing is repeated until a single digit remains, then
the original number is only divisible by 3 if this single digit is
divisible by 3, ie it is one of {3, 6, 9}
745 â†’ 7 + 4 + 5 = 16
16 + 1 + 6 = 7
7 is not divisible by 3 (it is not one of {3, 6, 9}), so 745 is not
divisible by 3
To be divisible by 5, the last digit must be 0 or 5
The last digit of 745 is 5 which is one of {0, 5}, so 745 is
divisible by 5
To be divisible by 9 the sum of the digits must also be divisible
by 9; if the summing is repeated until a single digit remains, then
the original number is only divisible by 9 if this single digit is
9 (otherwise this single digit gives the remainder when the
original number is divided by 9)
745 + 7 + 4 + 5 = 16
16 â†’ 1 + 6 = 7
7 is not 9, so 745 is not divisible by 9 (the remainder is 7)
To be divisible by 10 the last digit must be 0
The last digit of 745 is 5 which is not 0, so 745 is not divisible
by 10.
745 is not divisible by 2, 3, 9, 10
745 is divisible by 5. (MORE)

262144

In most classrooms, 60% is a failing grade.

111/1000
Yes:
0.111 = 111/1000 (the decimal terminates)
0.111... = 1/9 (the 1 recurs)

The LCM of 9 and 27 is 27.