Math

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How many integers between 200 and 500 inclusive leave a remainder 1 when divided by 7 and a remainder 3 when divided by 4?


Find the smallest positive integer that leaves a remainder 5 when divided by 7, a remainder 6 when divided by 11, and a remainder 4 when divided by 13.

  • Math -

    list multiples of 7 plus 1 past 200 :
    204 211 218 225 232 239 246..

    list multiples of 4 plus 3 past 200
    203 207 211 215 219 223 227 231 235 239 AHHHH

    239 is the smallest
    this will happen again in 28 (multiply the 2 remainders)
    so 239 267 295 323 ... 491
    number of terms ??
    consider it an arithmetic sequence where a = 239, d - 28 and n = ?
    t(n) = a+(n-1)d
    491 = 239 + (n-1)(28)
    252 = 28n - 28
    280 = 28n
    n = 10

    remainder of 5 when divided by 7 ---- 5 mod 7
    remainder of 6 when divided by 11 --- 6 mod 11
    remainder of 4 when divided by 13 --- 4 mod 13

    we could do the same thing:
    12 19 26 33 40 47 54 61 68 75 82 89 96 103 110 117 124 131 138 ...
    6 17 28 39 50 61 72 83 94 105 116 127 138 149...
    4 17 30 43 56 69 82 95 108 121 134 147...

    mmmhh?

    Let's try something completely different:
    "The Chinese Remainder Theorem"
    Google it to get several examples, there is a good Youtube

    Z = 5 mod 7
    Z = 6 mod 11
    Z = 4 mod 13

    X = 5b1 c1 + 6 b2 c2 + 4b3 c3
    to get b's, multiply the mods
    7x11x13 = 1001
    b1 = 1001/7 = 143
    b2 = 1001/11 = 91
    b3 = 1001/13 = 77

    sofar we have
    X = 5(143) c1 + 6(91)c2 + 4(77)c3

    now the tricky part
    143(c1) = 1 mod 7
    3c1 = 1 mod 7
    3c1 = 8mod7
    -4c1 = 8 mod 7
    c1 = -2

    91c2 = 1 mod 11
    3c2 = 1mod11
    3c2 = 12 mod 11
    c2 = 4

    77c3 = 1 mod 13
    12 c3 = 1 mod 13
    12c3 = -12 mod 13
    c3 = -1

    so X = 5(143)(-2) + 6(91)(4) + 4(77)(-1)
    = 446 mod 1001

    so the smallest such number is 446

    check: 446/7 = 63 remainder 5
    446/11 = 40 remainder 6
    446/13 = 34 remainder 4

    How about that, we could have gone for quite a bit using the simple method of just listing.

  • Math -

    446 divided by 11

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