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How many three-digit integers are multiples of both 4 and 6 and have a unit digit of 2?

  • math -

    to be multiples of 4 and 6, they must be multiples of 12, the LCM
    The smallest 3-digit multiple of 12 is
    108 , then 120 , 132 .. .996

    consider this to be an arithmetic sequence
    a = 108, d = 12 , n = ? -- the number of terms
    t(n) = a + (n-1)d
    996 = 108 + 12(n-1)
    888 = 12n - 24
    12n = 912
    n = 76

    There are 76 of them

  • math -

    Since 12n = 12(5m+1)= 60m +12, our goal is to now count integers m such that 100 less than or equal to 60m +12<1000. Subtracing 12 from each part of the inequality, we get 88 lees than or equal to 60m<988. Dividing by 60, we get 2 less than or equal to m less than or equal to 16, so there are 16-1=15 values for the integer m that give us the three-digit multiples of 4 and 6 that have units digits of 2.

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