A Guide to the GRE/Digits of Numbers

Digits of Numbers
If a question gives you the relationship between the various digits of a number, substitute variables for these digits and put the appropriate power of 10 in front of each one.

For example, the test might ask, “in a number, the hundreds digit is 2 greater than the tens digit, which is 2 greater than the units digit. If the number is equal to 195 multiplied by the units digit, what is the number?'


 * Let u equal the units digit, t the tens, h the hundreds, and x the number.
 * Restate the givens in terms of u, t, h, and x.
 * t = u + 2
 * h = t + 2 = u + 4
 * x = 195u
 * x = 100h + 10t + u
 * Find the value of u.
 * 195u = x = 100h + 10t + u
 * 194u = 100h + 10t
 * 194u = 100(u + 4) + 10(u + 2)
 * 194u = 100u + 10u + 420 = 110u + 420
 * 84u = 420
 * u = 420/84 = 5
 * Now you can find the values of the other numbers:
 * t = 5 + 2 = 7
 * h = 5 + 4 = 9
 * Check your work:
 * 975 = 195 * 5

Practice
1. The tens digit of a 3-digit number is twice the units digit and one less than the hundreds digit. If the number is equal to two less than 136 times the tens digit, what is the number?

2. If the digits of a 2-digit number are reversed, the resulting number is 45 greater than the original number. If the initial 2-digit number is less than 20, then what is its value?

Answers to Practice Questions
1. 542

Let t equal the tens digit.

100(t + 1) + 10t + t/2 = 136t - 2
 * Take the original equation.

100t + 100 + 10t + t/2 = 136t - 2
 * Expand the parentheses.

110.5t + 100 = 136t - 2
 * Combine variables and constants.

100 = 25.5t - 2
 * Subtract from both sides.

102 = 25.5t
 * Add 2 to both sides.

4 = t
 * Divide both sides by 25.5
 * t is equal to 4, and the number is thus equal to 542.

2. 16

Let t equal the tens digit and u equal the units digit.

10u + t - (10t + u) = 45		Take the equation.

10u + t - 10t - u = 45			Expand the parentheses.

9u - 9t = 45				Combine the variables.

u - t = 5				Divide both sides by 9.

u is 5 greater than t; thus, u and t could be 1 and 6, 2 and 7, 3 and 8, 4 and 9. However, since the initial value is less than 20, the digits must be 1 and 6, which make 16 and 64.