Date Multiplying Challenge (Inspired by last week's Riddler on FiveThirtyEight.com. Sandbox post.)
Given a year between 2001 and 2099, calculate and return the number of days  during that calendar year where mm * dd = yy (where yy is the 2-digit year).
2018, for example, has 5:

January 18th (1 * 18 = 18)
February 9th (2 * 9 = 18)
March 6th (3 * 6 = 18)
June 3rd (6 * 3 = 18)
September 2nd (9 * 2 = 18)

Input can be a 2 or 4-digit numeric year.
Output should be an integer. Optional trailing space or return is fine.
Complete input/output list:
Input = Output
 2001 = 1     2021 = 3     2041 = 0     2061 = 0     2081 = 2
 2002 = 2     2022 = 3     2042 = 4     2062 = 0     2082 = 0
 2003 = 2     2023 = 1     2043 = 0     2063 = 3     2083 = 0
 2004 = 3     2024 = 7     2044 = 3     2064 = 2     2084 = 5
 2005 = 2     2025 = 2     2045 = 3     2065 = 1     2085 = 1
 2006 = 4     2026 = 2     2046 = 1     2066 = 3     2086 = 0
 2007 = 2     2027 = 3     2047 = 0     2067 = 0     2087 = 1
 2008 = 4     2028 = 4     2048 = 6     2068 = 1     2088 = 3
 2009 = 3     2029 = 1     2049 = 1     2069 = 1     2089 = 0
 2010 = 4     2030 = 6     2050 = 3     2070 = 3     2090 = 5
 2011 = 2     2031 = 1     2051 = 1     2071 = 0     2091 = 1
 2012 = 6     2032 = 3     2052 = 2     2072 = 6     2092 = 1
 2013 = 1     2033 = 2     2053 = 0     2073 = 0     2093 = 1
 2014 = 3     2034 = 1     2054 = 4     2074 = 0     2094 = 0
 2015 = 3     2035 = 2     2055 = 2     2075 = 2     2095 = 1
 2016 = 4     2036 = 6     2056 = 4     2076 = 1     2096 = 4
 2017 = 1     2037 = 0     2057 = 1     2077 = 2     2097 = 0
 2018 = 5     2038 = 1     2058 = 0     2078 = 2     2098 = 1
 2019 = 1     2039 = 1     2059 = 0     2079 = 0     2099 = 2
 2020 = 5     2040 = 5     2060 = 6     2080 = 4

This is a code-golf challenge, lowest byte count in each language wins.
Pre-calculating and simply looking up the answers is normally excluded per our loophole rules, but I'm explicitly allowing it for this challenge. It allows for some interesting alternate strategies, although its not likely a 98 99-item lookup list is going to be shortest.
Excel, 48 bytes

Hooray! Finally something Excel is actually good at. 

=COUNT(VALUE(ROW(1:12)&amp;"/"&amp;A1/ROW(1:12)&amp;"/"&amp;A1))


Takes input from A1 in the form of an integer 1-99 representing the year, and outputs to wherever you enter this formula. It's an array formula, so use Ctrl-Shift-Enter instead of Enter to enter it. 

This takes advantage of the fact that COUNT ignores errors, so any errors that are caused by either the month not dividing the year (leading Excel to parse something like 2/12.5/25 or by the date not being valid, like 2/29/58, are just silently ignored.