Another Math-gic Trick

Number Theory Level pending

Bob became bored again, so he asked his friend to tell him a trick again. The friend said:

"Okay. Hm............. okay. Think of a number between 1 and 1000. Now, divide it by 7 then tell me the remainder."

Bob replied 4.

"Now divide it by 11 then tell me the remainder."

Bob replied 1.

"Now divide it by 13 then tell me the remainder."

Bob replied 8.

"Okay, I now know your number, it's......."

What is the number Bob was thinking of?

2 solutions

Sunil Pradhan
Mar 25, 2014

N0 = 7a + 4 = 11b + 1 = 13c + 8

7a = 11b – 3 = 13c + 4 using 11b – 3 = 13c + 4

b = (13c + 7)/11by trial and error c = 2, 13, 24, ... find number and find remainder when divided by 7

46 when c = 46

No . 46 × 13 = 8 = 606

Sarath Ch
Jan 22, 2014

if n is the number then it will be of the form 13x+8.

now n when divided by 7 gives 4 as remainder.

so 13x when divide by 7 remainder is 3.

so 13x = 14x-x = 3(mod 7).

=> x = 4(mod7).

so x = 7p+4.

so n = 13(7p+4)+8 = 91p + 60.

and n when divide by 11 remainder is 1.

so 91p +60 = 1(mod 11).

so 91p = 7(mod 11) => 3p = 7(mod 7) which is true for p = 6.

so n= 91p+60 = 540 + 60 = 606.