A number when divided by 28, leaves a remainder of 7. How do you find the remainder when the same number is divided by 35?

Question

Hazel Anglin · Accepted Answer

Sometimes, in attempting to answer a question about numbers, it is helpful to look at some examples.

Below are the first 11 (positive) numbers that leave a remainder of #7# when divided by #28#. Note that the description implies that the numbers are of the from #28n+7# for some (positive) integer #n#. Look at the first three columns: #{:(bb" n ",bb" 28n+7 ",bb" 35q+r ",bb" R(n/5)"), (" "0," "" "7,35xx0+color(red)(7)," "" "0), (" "1," "" "35,35xx1+color(red)(0)," "" "1), (" "2," "" "63,35xx1+color(red)(28)," "" "2), (" "3," "" "91,35xx2+color(red)(21)," "" "3), (" "4," "" "119,35xx3+color(red)(14)," "" "4), (" "5," "" "147,35xx5+color(red)(7)," "" "0), (" "6," "" "175,35xx5+color(red)(0)," "" "1), (" "7," "" "203,35xx6+color(red)(28)," "" "2), (" "8," "" "231,35xx7+color(red)(21)," "" "3), (" "9," "" "259,35xx8+color(red)(14)," "" "4), (" "10," "" "287,35xx9+color(red)(7)," "" "0) :}# First of all, it is now clear that knowing only the remainder when a number is divided by #28# is not enough to uniquely determine the remainder when it is divided by #35#. But there is a pattern. So we don't need complete information about the original number or about #n#. (If we know one of those, we could find the other.) The pattern has a cycle of 5 values, so it makes sense to look at the fourth column, the remainder when #n# is divided by #5#. Writing the result and proving it for all possible #n# is left to the reader.

Savannah Anderson · Answer

undefined To find the remainder when a number is divided by 35, we need to consider the common factors of 28 and 35, which is 7. Since the remainder when the number is divided by 28 is 7, and 28 is a multiple of 7, the remainder when the same number is divided by 35 will also be 7.