A number divided by 12 leaves remainder 7. what will be the remainder if the sane number is divided by 6
Answers
Answer:
Remainder when divided by 7; number is 18a+7 for some integer a.
Similarly, the same is 12b+n for some integer b.
So,
18a+7=12b+n
<=>18a-12b=n-7
<=>6(3a-2b)=n-7
(3a-2b) is integer. So, 6(3a-2b) is integer multiple of 6.
Hence, n-7 is multiple (need not be positive) of 6 and n is between 0 and 11 (by definition of remainder).
Such values of n are 1 and 7.
Ans: Values are 1 and 7; Number of values 2.
Alternate method:
If we divide a number by divisor by d, then reminder of number after adding multiple od d in it or subtracting multiples d from leaves the same remainder. This can be easily proven.
For given number, we can add or subtract multiple of 36 such that new number is between 0 and 35 (simple, remainder when divided by 36) which will have the same remainder as when we divide by 18.
Similarly, the same number the same remainder when that when we divide by 12.
The numbers from 0 to 35 leaving remainder 7 are 7 and 25.
Remainder for these when divided by 12 are 7 and 1 respectively.
Ans: Values are 1 and 7; Number of values 2.
One more method:
The number leaves remainder 7 when divided by 18. So, it leaves remainder 7 or 25 when divided by 36.
Hence, it leaves remainder 7 or 1 when divided by 12.
Ans: Values are 1 and 7; Number of values 2.