Question

x is least number divided by 5,6,8,9,12 leaves remainder 1 but divisible by 13

Answer 1

suppose we are given number when divided by m,y,z ;it leaves remainder a,b,c. Then the number format will be "LCM(m,y,z)*x+constant".

the question is find the value of constant.If all  of them leave the same remainder 'r', constant = r. It can also be   looked at as the smallest number satisfying the given property.

So the number N=LCM(5,6,8,9,12)*x+1= 360x+1

now we have to find out suitable value of x that 360x+1 is divisible by 13. we know that 351 is divisible by 13.

So, 351x+9x+1 is a multiple of 13

now we have to find x for which 9x+ 1 is divisible by 13. that is, 9*10+1=91

the least number is 360*10+1=3601

Answer 2

$\huge \boxed{ \underline{ \underline{ \bf{Answer}}}}$

FOR FINDING THE NUMBER WE SHOULD HAVE TO FIRST FIND L.C.M OF ALL :-

L.C.M Of  5,6,8,9,12  = 360

Now 360 × 1+ 1 = 361 which is not divisible by 13.

360 × 2 + 1 = 721  which is not divisible by 13.

360 × 3 + 1 = 1081 which is not divisible by 13.

360 × 4 + 1 = 1441 which is not divisible by 13.

360 × 5 + 1 =  1801 which is not divisible by 13.

360 × 6 + 1 = 2161 which is not divisible by 13.

360 × 7 + 1 = 2521 which is not divisible by 13.

360 × 8 + 1 = 2881 which is not divisible by 13.

360 × 9 + 1 = 3241 which is not divisible by 13.

360 × 10 + 1 = 3601,  which is divisible by 13.

$\implies \boxed{\mathsf{Therefore,\:Required\:Number\:=3601.}}$