## Odd Digit Sums

Let \(N! = 1 \times 2 \times 3 \times \cdots \times N\).
A positive integer is called *valid* if the sum of its digits is an odd number.
For example \(131\) is valid because \(1+3+1=5\) and \(5\) is odd.
For any positive integer \(x\) let \(v(x)\) denote the number of valid positive integers which are divisors of \(x\) (including \(1\) and \(x\) itself).
For example \(v(100) = 6\) because \(100\) has \(6\) valid divisors: \(\{1, 5, 10, 25, 50, 100\}\).

Part A | Find \(v(720)\). | |
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Part B | Find \(v(10!)\). | |

Part C | Find \(v(26!)\). |

## Counting Triangles

For each positive integer \(N\), let \(t(N)\) denote the number of right angled triangles with perimeter less than or equal to \(N\) having positive integer side lengths. For example \(t(30)=3\) with the three examples of right angled integer triangles being: \((3,4,5)\), \((6,8,10)\) and \((5,12,13)\).

Part A | Determine \(t(666)\) | |
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Part B | Determine \(t(666666)\) |