# Find how many 3 digits numbers satisfy all the following conditionsif it is divided by 2, the remainder is 1,if it is divided by 3, the remainder is 2,if it is divided by 4, the remainder is 3,if it is divided by 5, the remainder is 4,if it is divided by 8, the remainder is 7,

$let \;the \;number \;N \;satisfy \;the\;conditions.\\* Let \;x, \;y,\; z,\; w, \;v \;be \;some \;integers.\\* N\;can\;be\;written\;as\;2x+1\; or\; 2x -1.\* so\; let \;us\; write \;N \;as\\* \Rightarrow N = 2 x - 1\\* \Rightarrow N = 3 y - 1\\* \Rightarrow N = 4 z - 1\\* \Rightarrow N = 5 w - 1\\* \Rightarrow N = 8 v - 1\\* So ,\;N + 1 = 2 x = 3 y = 4 z = 5 w = 8 v\\* N+1 \;is \;a\; multiple \;of \;2,\;3,\;4,\;5, \;and \;8\\* N+1 \;is\; a \;multiple \;of \;LCM \;of\; them.\\* LCM=2\times 3\times 2\times 5\times 2=120\\* Three \;digit\; multiples \;of \;120 \;are \;120, \;240,\; 360,\; 480, \;600, \;720,\; 840,\; 960\\* N = 119,\; 239, \;359,\; 479, \;599,\; 719,\; 839, \;959\\* There \;are \;eight\;such\; three\;digit\;numbers.$

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