A natural number N when successively divided by 4,5 and 6 leaves remainder 2, 4 and 5 respectively. What is the sum of the remainders obtained when N is successively divided by 12 and 10?
Let N = another natural number + 2 = 4a + 2 = 5b + 4 = 6c + 5.
4a - 2 = 5b, so a - b = 3 (let's call this equation I).
4(a - b) = 20.
Now we have 4a + 2 = 5b + 20 (let's call this equation II).
So 5b - 5b - 20 = 2 - 4.
10 = 2c.
c = 5.
10 is divisible by both 10 and 12, so the sum is <<10+12=22>>22. Answer: \boxed{22}.
To find the sum of the remainders obtained when N is successively divided by 12 and 10, we need to first find the value of N.
Given that N leaves remainder 2 when divided by 4, 4k + 2 = N (where k is a positive integer)
N leaves remainder 4 when divided by 5, 5m + 4 = N (where m is a positive integer)
N leaves remainder 5 when divided by 6, 6n + 5 = N (where n is a positive integer)
To find the value of N, we can equate these three equations and solve for N.
4k + 2 = 5m + 4
5m + 4 = 6n + 5
Simplifying the equations:
4k - 5m = 2
5m - 6n = 1
Now, let's find the value of k, m, and n:
From the first equation, let k = 5p + 1 (where p is a positive integer)
Substituting k into the second equation:
5m - 6n = 1
5(5p + 1) - 6n = 1
25p + 5 - 6n = 1
25p - 6n = -4
Simplifying the equation:
25p - 6n = -4
To solve this equation, we need to find the values of p and n that satisfy the equation. Let's try different values of p and n:
p = 1, n = 4:
25(1) - 6(4) = -1 ≠ -4
p = 2, n = 9:
25(2) - 6(9) = 50 - 54 = -4
So, p = 2 and n = 9 satisfy the equation. Substituting these values into the equation for k:
k = 5p + 1
k = 5(2) + 1
k = 10 + 1
k = 11
Now, we can find the value of N using any of the given equations. Let's use the equation N = 4k + 2:
N = 4(11) + 2
N = 44 + 2
N = 46
So, N is equal to 46.
To find the sum of the remainders obtained when N is successively divided by 12 and 10:
Remainder when N is divided by 12 = N % 12 = 46 % 12 = 10
Remainder when N is divided by 10 = N % 10 = 46 % 10 = 6
Therefore, the sum of the remainders obtained when N is successively divided by 12 and 10 is 10 + 6 = 16.