Remainders

If the remainder is 7 when positive integer n is divided by 18, what is the remainder when n is divided by 6?

n = 18q + 7 Now, since the first term (18q) is divisible by 6, then the remainder will only be from the second term, which is 7. 7 divided by 6 yields the remainder of 1. 18g/6 + 7/6 = 3q+7/6

If n divided by 7 has a remainder of 2, what is the remainder when 3 times n is divided by 7?

n = 7q+2 3n = 3(7q)+6 6 is a remainder

n divided by 3 gives a remainder 1 and n divided by 7 gives a remainder 5. What can we say?

n=3a+1 n=7b+5 n makes 3 group with 1 leftover and n makes 7 group with 5 leftover. In both cases we need 2 to make additional group. Therefore, we can say n = 3x - 2 n = 7y - 2 and 2 is common remainder.

Pay attention to division by 10, 100, 1000, ...

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The sum of the digits can be used to find the remainder.

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If k divided by 5 gives the remainder 1 then k can be equal?

1 as 1/5=0 (1/5) with the remainder of 1 and quotient 0

If we have a number N, which when divided by 3 gives a remainder 1 and when divided by 7 gives a remainder 1, what would be the remainder when N is divided by 21?

1 as well 3 + 1, 6 + 1, 9+ 1, 12 + 1, 15 + 1, 18 + 1, 21 + 1 etc 7 + 1, 14 + 1, 21 + 1, 28 + 1 etc

What are 4 patterns of remainder problems when algebraic approach works better?

1) Ratios. s/t = 64.12. CROSS-MULTIPLY 2) Similar remainders. FIND COMMON REMAINDER by SUBTRACTION. TEST 3) Different remainders. FIND LCM (before q) and TEST; KEEP ADDING LCM 4) With exponents. TEST remainders for ODD/EVEN exponents, e.g. 51^25/13 - remainder is 12 as 25 is odd For other problems use plug-in method

What is the remainder when p/4 if p = 8q+5?

p = 8q+5 p = (8q+4)+1 p = 4(2q+1)+1 so the remainder upon division of p by 4 is 1

D/S = 1876.375. What is the remainder?

r/S = 0.375 = 3/8 = 6/16, etc. thus, the proportion, but not exact numbers, is found

s/t = 64.12 What could be a remainder? 2, 4, 8, 20, 45

r/t = 12/100 25r = 3t -> t = 25r/3 r is a multiple of 3, so only 45 suits

Express 10/n has a remainder of 4

10=qn+4 D/S = Q + r/S -> D = S*Q + r

What numbers, when divided by 12, have a remainder of 5?

12+5=17 (q=1) 24+5=29 (q=2) 36+5=41 (...) ... keep adding 12

If 13,333 - n is divisible by 11 what is the remainder?

13333 - 11000 = 2333 2333 - 2200 = 133 133 - 121 = 12 12 - 11 = 1

What does it mean 2(x+y) = 5x?

2(x+y) is divisible by 5, so too is (x+y)

What is the smallest positive integer that, when divided by 12, has a remainder of 5?

5 as 5/12 has the quotient 0 and the remainder 5

How to find number of values for integer n when remainder of two numbers are not equal to form complete groups?

Find LCM of factors, test for reminders to find n, keep adding LCM n = 8a + 5 n = 13b + 2 LCM = 104 If b = 7, n = 93. Is it of the form (8a + 5)? Yes! n/8 (or 93/8) gives a remainder of 5. The next value of n = 93 + 104 = 197

If you divide 7^131 by 5, which remainder do you get?

cyclicity of 7 = 4 so when we divide 131 by 4 we get 3 as remainder therefore 7^3 = 343 now divide 343 by 5 and u get the remainder as 3