Question

. Find the least multiple of 7 which when divided by 2, 3, 4, 5
and 6 leaves the remainders 1, 2,3, 4 and 5 respectively.​

Answer 1

Answer:

Yes, 119 is the correct answer, but why?

The simplest way to look at this is by starting with the rem 4 when divided by 5 fact. This means that the answer is among these numbers: 4, 9, 14, 19, 24, ......

Also, the rem 1 when divided by 2 means the number is odd, so scratch all those above that end in 4,

9, 19, 29, 39,......

Notice that 9 and 29 leave a remainder of 1 when divided by 4.

This means they aren't it, so that leaves: 19, 39, 59,.....

Now, notice that we need a remainder of 2 when divided by 3.

This means that the digit sum of the number must be 2.

1 + 9 = 10 = 1 no

3 + 9 = 12 = 0 no

5 + 9 = 14 => 1 + 4 = 5 => 2 yes

......

So this means that we start with 59, add multiples of 20, and add the digits. When we get to 2, we divide by 7. The first one we get is the answer.

59 / 7 no

79 / 7 no

99 / 7 no

119 / 7 yes

After only 4 tries, we have found it.

Actually, using the Chinese Remainder Theorem in number theory, we don't need to guess, but that requires knowledge of modular arithmetic, so not everybody would get that.

hope this will help you