Number Bases in CAT: 3 Conversion Rules That Cover Most

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Number bases look intimidating the first time you see a question written as (324) in base 5. The notation is the whole difficulty. Once you read the digits as a short sum of powers, most of these questions turn into ordinary arithmetic, and the rest turn into one small equation.

This guide gives three conversion rules that cover most of what CAT asks on bases, then shows how to add and multiply inside a base and how to handle questions where the base itself is the unknown. Every worked example below has been checked by converting back to decimal, which is also the habit I want you to build.

What a Base Actually Means

In base 10 the number 347 means 3 hundreds, 4 tens and 7 ones. That is 3 times 10 squared, plus 4 times 10, plus 7. Every base works the same way. In base 5, the digits can only be 0 to 4, and each place is worth 5 times the place to its right. So (324) in base 5 means 3 times 25, plus 2 times 5, plus 4.

The main point is that the digits never reach the base. A digit 5 cannot exist in base 5, just as a digit 10 cannot exist in base 10. If a question shows a digit equal to or larger than the stated base, either the base is wrong or you have found the clue that tells you the base must be larger than that digit.

Most of the exam value in this chapter comes from one habit. Whenever you are unsure, write the place values above the digits and expand. It takes ten seconds and removes the notation from the problem.

Rule 1: Expand to Decimal

To convert from base b to decimal, multiply each digit by its place value and add. For (324) in base 5, the places are 25, 5 and 1, so the value is 3 times 25 plus 2 times 5 plus 4, which is 75 plus 10 plus 4, or 89.

For longer numbers, work from the left with the doubling style recurrence: take the first digit, multiply by the base, add the next digit, and repeat. For (324) in base 5 that is 3, then 3 times 5 plus 2 equals 17, then 17 times 5 plus 4 equals 89. You get the same answer with fewer powers to remember, and it is the cleaner method for numbers with five or six digits.

This rule is also how you read a question stated in another base. If the question gives numbers in base 7 and asks for a comparison, converting each to decimal first is a safe, if slow, route. The faster routes below are for when you recognise the structure.

Rule 2: Repeated Division for Decimal to Base b

To convert a decimal number into base b, divide by b, write down the remainder, divide the quotient by b again, and continue until the quotient is zero. The remainders, read from the last to the first, are the digits.

Take 157 and convert it to base 6. Dividing 157 by 6 gives quotient 26 and remainder 1. Dividing 26 by 6 gives quotient 4 and remainder 2. Dividing 4 by 6 gives quotient 0 and remainder 4. Reading the remainders from the bottom up gives (421) in base 6. Check by expanding: 4 times 36 is 144, plus 2 times 6 is 12, plus 1, which gives 157. The check works, so the answer stands.

Notice what the first remainder is. The last digit of a number in base b is always its remainder on division by b, so the units digit of 157 in base 6 is 1 because 157 leaves remainder 1 on division by 6. That one fact answers a whole family of questions about last digits in other bases without any conversion at all.

Task	Method	Worked example
Base b to decimal	Expand with place values	(324) in base 5 equals 89
Decimal to base b	Divide repeatedly, read remainders upward	157 equals (421) in base 6
Last digit in base b	Remainder on division by b	157 ends in 1 in base 6
Number of digits in base b	Find the smallest power of b above the number	1000 has 10 digits in base 2

Rule 3: Group Digits for Related Bases

When one base is a power of another, you do not need decimal at all. Base 8 is 2 cubed, so three binary digits make one octal digit. Base 16 is 2 to the fourth, so four binary digits make one hexadecimal digit.

Take the binary number 110101101. To go to octal, group from the right in threes: 110, 101, 101. Each group is a digit: 6, 5, 5. So the number is (655) in base 8. To go to base 16, group in fours from the right and pad the left with zeros: 0001, 1010, 1101. Those are 1, 10 and 13, which in hexadecimal are 1, A and D, giving (1AD). As a check, all three forms equal 429 in decimal.

CAT questions on bases are usually in small bases such as 3, 4, 5, 6 and 7, so grouping appears less often than rules 1 and 2. When it does appear, it saves a long conversion. If a question gives a binary number and asks for its octal digits, grouping is the intended route.

Adding and Multiplying Inside a Base

You can do arithmetic directly in a base, carrying whenever a column reaches the base instead of 10. Add (234) and (143) in base 5. The units column is 4 plus 3, which is 7. In base 5, 7 is one five and two ones, so write 2 and carry 1. The next column is 3 plus 4 plus the carry 1, which is 8, or one five and three ones, so write 3 and carry 1. The last column is 2 plus 1 plus 1, which is 4. The answer is (432) in base 5.

Check it in decimal. (234) in base 5 is 69, (143) in base 5 is 48, and their sum is 117. And (432) in base 5 is 4 times 25 plus 3 times 5 plus 2, which is 117. The numbers match.

Subtraction works the same way with borrowing, where a borrowed unit is worth the base and not 10. In a hurry, converting to decimal, doing the arithmetic, and converting back is often the safer path. Use direct arithmetic when the numbers are short and the base is small.

When the Base Is the Unknown

Some questions give you a number written in an unknown base b and ask you to find b. Write the number as a sum of powers of b, and set it equal to the other side. If the number 144 in base b equals 100 in decimal, then b squared plus 4b plus 4 equals 100. That is (b plus 2) squared equals 100, so b plus 2 equals 10 and b equals 8. The digit 4 is smaller than 8, so the base is legal.

The legality check matters. An equation can have a root that breaks the digit rule, and the digit rule is how a question writer tells you which root to keep. If you find two possible bases, the larger digit in the problem rules out the smaller one.

Another common form gives an operation done in the unknown base, such as a sum that comes out as a round number. Convert every term to an expression in b, form the equation, and solve. These usually reduce to a quadratic, which you can solve by factorising, so keep your equation skills close. Our guide on quadratic equations for CAT covers the factorising habits.

Counting Digits and Place Values

To count the digits of a number N in base b, find the smallest power of b that is bigger than N. If b to the power k is the first such power, the number has k digits. For 1000 in base 2, 2 to the 9 is 512 and 2 to the 10 is 1024, so the number has 10 digits.

This helps with questions such as how many digits a power has in another base. Because the digits count rises each time the number passes a power of the base, it behaves like a logarithm. If the question is about very large powers, you may need logarithms. Our guide on the four recall tables for mental math gives you the powers of small numbers that make this quick.

Place value questions also appear in a disguised form: a two digit number in base b whose digits swap and give a related number. Write the number as xb plus y, write the swapped number as yb plus x, and use the stated relation. The algebra is the same as for base 10 digit problems, with b in place of 10.

When the rules feel automatic, move on to timed sets, and keep your formula notes in one place on the CAT cheatsheets page. If you want a second view on how much time bases deserve in your plan, you can talk to a CAT mentor.